Arguably, the two most notable British Unitarians who influenced the American Founding were the 
Socinian Joseph Priestley and 
Arian Richard Price.
Thomas Jefferson, self proclaimed unitarian, militantly so, professed great admiration for both (theologically he was closer to Priestley; though arguably Jefferson was even more heterodox than Priestley. I'm assuming that Socinianism is more heterodox than Arianism).
Some controversy ensues over whether Jefferson was heterodox before 1813. Every serious Jefferson scholar thinks he was. See Warren Throckmorton's ongoing case, all the details of which admittedly, I haven't memorized.
I know Throckmorton invokes a 1788 letter by Jefferson on refusal to be a Godfather because TJ didn't want to affirm the Trinity. 
There Jefferson notes a lifelong "difficulty" with that doctrine. Throckmorton offers other evidence too, noting:
Jefferson also confided to a Unitarian friend that he attended 
Priestley’s Unitarian church before 1800, while he was Vice President. 
In Jefferson’s 1803 Syllabus, he laid out his belief that Jesus was not 
part of the Godhead. [The] attempt to make Jefferson seem orthodox 
during the active part of his political engagement is contradicted by 
Jefferson’ own words.
Throckmorton may have included perhaps (?) what I disclose below. But I want to re-fresh the record.
Let's start backwards with where Jefferson ended. On 
February 27, 1821, writing to Timothy Pickering, Jefferson stated:
I do not know that you and I may think alike on all points. as the 
Creator has made no two faces alike, so no two minds, and probably no 
two creeds. we well know that among Unitarians themselves there are 
strong shades of difference, as between Doctors Price and Priestley for 
example. so there may be peculiarities in your creed and in mine. they 
are honestly formed without doubt. I do not wish to trouble the world 
with mine, nor to be troubled for them. these accounts are to be settled
 only with him who made us; and to him we leave it, with charity for all
 others, of whom also he is the only rightful and competent judge. I 
have little doubt that the whole of our country will soon be rallied to 
the Unity of the Creator, and, I hope, to the pure doctrines of Jesus 
also.
The context of this letter has Jefferson railing against "the incomprehensible jargon of the Trinitarian arithmetic, that three are one, and one is three," while granting latitude towards the Arianism and Socinianism of Price and Priestley. 
It's hard for anyone to dispute Jefferson from 1813 onward held to such a heterodox unitarian position. 1813 seemed a watershed year for both Jefferson AND J. Adams where they bitterly rejected and mocked the Trinity. That was the year Great Britain finally got the law off its books making it a crime to publicly deny the doctrine. When reading 
their correspondence, skip to 1813. That year is practically all about their rejection of the Trinity and orthodox Christian doctrine. 
So then let's go to the beginning. The earliest I trace Jefferson's connection to Priestley and Price is the mid 1780s, after the Declaration of Independence, but before the Constitution was ratified and Jefferson's subsequent service in the newly formed Federal government.
Jerry Newcombe and  Mark A. Beliles have a new book out which explores the current controversy (and examines the arguments of among others, David Barton, Gregg Frazer, Throckmorton and his co-author Michael Coulter). Though known to sympathize with the "Christian America" view of history, the timeline they give 
on page 380 seems accurate enough (to my eyes). I would stress, though, there is not sufficient evidence that Jefferson's turn towards explicit unitarianism at that time -- 1785-86 -- came from the orthodox Trinitarian direction as opposed to one more deistic and less self consciously "Christian" than where Jefferson's beliefs terminated. 
Newcombe and Beliles document how John Adams, meeting with Jefferson in England in 1786 took him to a Unitarian Church with a service led by Richard Price.  
Jefferson penned a letter to Price in 1785 praising Price's 
Observations on the American Revolution which Price earlier sent a copy. Price also sent a copy to among others, 
George Washington who similarly praised the tract. Price's addresses is explicitly pro-unitarian.
In their ensuing correspondence, Price further discusses religion with Jefferson. 
As Price wrote him on October 26, 1788:
I am now reading Mr. Necker’s book on the importance of religious 
opinions. ...  He should have defined it, and taken care 
to distinguish the religion he means from the Superstitions that go 
under the name of religion, and which have done unspeakable harm in the 
world. What he Says is true only of a rational and liberal religion; 
that is of a religion which enforces the obligations of morality by 
motives drawn from the authority of a righteous and benevolent Deity and
 a future retribution. But he Seems never to have consider’d that there 
has been in almost all religions a melancholy Separation of religion 
from morality. Popery teaches a method of pleasing God without forsaking
 vice, and of getting to heaven by penances, bodily mortifications, 
pilgrimages, saying masses, believing mysterious doctrines, burning 
heretics, aggrandizing Priests &c. Mahometans expect a paradise of 
Sensual pleasures. Pagans worship’d lewd, revengeful and cruel Deities, 
and thus Sanctify’d to themselves1
 Some of the worst passions. The religion likewise of many Protestants 
is little better than a compromise with the Deity for wrong practises by
 fastings, Sacraments hearing the word &c. Would not Society be 
better without Such religions? Is Atheism less pernicious than Demonism?
 And what is the religion of many persons but a kind of demonism that 
delights in human Sacrifices and causes them to look with horror on the 
greatest part of mankind? Plutarch, it is well known, has observd very 
justly that it is better not to believe in a God than to believe him to 
be a capricious and malevolent being. These reflexions have Struck me 
very forcibly in reading Mr. Necker’s book. They shew how incumbent it 
is on all who wish the happiness of the world to endeavour to propagate 
just notions of the Deity and of religion. I can reflect with Some 
Satisfaction that this has been one of the Studies and labours of my 
life.
Jefferson's response, dated January 8 of that year, affirms Price's sentiment while connecting their shared enlightened heterodox theological zeitgeist to the US Constitution.
I was favored with your letter of October 26th[;]
... its subjects ... were to me, as everything which comes from you,
pleasing and instructive.  I concur with you strictly in your opinion
of the comparative merits of atheism and demonism, and really see
nothing but the latter in the being worshipped by many who think
themselves Christians.  Your opinions and writings will have effect
in bringing others to reason on this subject.  Our new Constitution,
of which you speak also, has succeeded beyond what I apprehended it
would have done. 
Then in a letter written 
on July 12, 1789, Jefferson asks Price the following:
Is there any thing good on the subject of the Socinian doctrine, 
levelled to a mind not habituated to abstract reasoning? I would thank 
you to recommend such a work to me. Or have you written any thing of 
that kind? That is what I should like best, as none are so easy to be 
understood as those who understand themselves. 
Price replies, August 3, 1789 introducing Jefferson to Joseph Priestley's Socinian writings:
In consequence of your desire that I would convey to you some tracts on 
the Socinian doctrine, I desire your acceptance of the volume of Sermons
 and the pamphlets that accompany this letter. The first part of Dr. 
Priestley’s letters I cannot immediately get; but it shall be sent to 
you by the first opportunity. The pamphlet entitled Two Schemes of a Trinity &c. is reckoned by the Socinians one of the best of all the 
publications in favour of their doctrine. You will see that Dr. 
Priestley and I differ much, but we do it with perfect respect for one 
another. He is a materialist and fatalist and we published some years 
ago a correspondence on these Subjects. ...
Jefferson then develops a fascination with Joseph Priestley's writings. It's evident that from 1813 onward Priestley had become Jefferson's favorite theologian. Jefferson corresponded with Priestley in the window between when Price introduces Jefferson to Priestley's writings and 1813. In a later post I may detail more on Jefferson's pre-1813 correspondence with Priestley. 
Rather, let's examine how Jefferson invokes both Price and Priestley in the year 1800. 
Writing to an ideological confidant, Bishop James Madison, on January 31, 1800, Jefferson praises Adam Weishaupt of the Illuminated Freemasonry legend. Before we see what Jefferson wrote, I note I don't see this as part of any kind of nefarious conspiracy. Freemasonry at the time was theistic, virtue orientated and religiously ecumenical in a way that was in principle universalistic. Hence it "fit" with their enlightenment zeitgeist.
The Illuminated Masonry of Weishaupt came to be known as the stuff of conspiracy and legend. But the context of Jefferson's letter, as I see it, has Jefferson trying to shoehorn Weishaupt's theology into his projected ideal of the works oriented unitarian theologies of Price and Priestley.
Wishaupt seems to be an enthusiastic Philanthropist. he is among those 
(as you know the excellent Price and Priestly also are) who believe in 
the indefinite perfectibility of man. he thinks he may in time be 
rendered so perfect that he will be able to govern himself in every 
circumstance so as to injure none, to do all the good he can, to leave 
government no occasion to exercise their powers over him, & of 
course to render political government useless. this, you know is 
Godwin’s doctrine, and this is what Robinson, Barruel & Morse
 have called a conspiracy against all government. Wishaupt believes that
 to promote this perfection of the human character was the object of 
Jesus Christ. that his intention was simply to reinstate natural 
religion, & by diffusing the light of his morality, to teach us to 
govern ourselves. his precepts are the love of god & love of our 
neighbor. and by teaching innocence of conduct, he expected to place men
 in their natural state of liberty & equality. he says, no one ever 
laid a surer foundation for liberty than our grand master, Jesus of 
Nazareth. he believes the Freemasons were originally possessed of the 
true principles & object of Christianity, and have still preserved 
some of them by tradition, but much disfigured. the means he proposes to
 effect this improvement of human nature are ‘to enlighten men, to 
correct their morals & inspire them with benevolence. secure of our 
success, sais he, we abstain from violent commotions. to have foreseen 
the happiness of posterity & to have prepared it by irreproacheable 
means, suffices for our felicity. this tranquility of our consciences is
 not troubled by the reproach of aiming at the ruin or overthrow of 
states or thrones.’ as Wishaupt lived under the tyranny of a despot 
& priests, he knew that caution was necessary even in spreading 
information, and the principles of pure morality. he proposed therefore 
to lead the Freemasons to adopt this object, and to make the objects of 
their institution, the diffusion of1
 science & virtue. he proposed to initiate new members into this 
body by gradations proportioned to his fears of the thunderbolts of 
tyranny. this has given an air of mystery to his views, was the 
foundation of his banishment & the subversion of the Masonic order, 
and is the colour for the ravings against him of  Robinson,
 Barruel & Morse, whose real fears are that the craft would be 
endangered by the spreading of information reason & natural morality
 among men.—this subject being new to me, I have imagined that if it be 
so to you also, you may recieve the same satisfaction in seeing, which I
 have had in forming the Analysis of it: and I believe you will think 
with me that if Wishaupt had written here, where no secrecy is necessary
 in our endeavors to render men wise & virtuous, he would not have 
thought of any secret machinery for that purpose: as Godwin, if he had 
written in Germany, might probably also have thought secrecy & 
mystycism prudent.
Now, this letter was written in 1800, just before Jefferson became President of the United States. Though not railing against the Trinity, he seems pretty clearly in Price and Priestley's unitarian theological camp, expositing heterodox political theology.
One final thing for now. It's my contention that the self consciously "unitarian" "Christianity" of the 1813 Jefferson onward was arrived at NOT from an orthodox Trinitarian direction, but rather the other side. From a place more deistic and less self consciously "Christian."
As I have noted before, I think Jefferson's chief theological influence before encountering Price, Priestley (and Conyers Middleton, who also cut up a Bible and was 
a source TJ named in 1813 with Priestley), was the more deistic 
Lord Viscount Bolingbroke. 
If Jefferson were coming from an orthodox Trinitarian direction towards his ultimate destination, why didn't he apparently stop at Price's Arianism for a short while? What I observe is Jefferson becoming familiar with and praising Price's works oriented unitarianism in 1785 and by 1789 asking about Socinianism and consequently being introduced to Priestley to whom Jefferson seems immediately to appreciate.
Likewise, the place where Jefferson from 1813 onward ends up is MORE heterodox than Priestley's Socinianism. As I have noted before, Priestley never (as far as I know) like Jefferson did, cut out large portions of the Bible -- like everything St. Paul "revealed" -- as false. And Priestley unlike Jefferson affirms Jesus' resurrection.  
Below is how I envision a theological spectrum from most orthodox to most heterodox:
1. Christian orthodoxy; 2. Arianism; 3. Socinianism; 4. Deism; 5. Atheism. 
I know reality can be more complicated and not so easily boxed. I don't think Jefferson was ever an atheist. And the "Deism" of 4 doesn't necessarily equate with cold "strict deism." But looking at the larger picture I see Jefferson, in his adult life moving from Bolingbroke's Deism towards Priestley's Socinianism. But remaining stuck in the middle. 
In short, I'd rate Jefferson a 3.5. (Likewise, since "Deism" is a broader category than what we may previously have thought, the cold, strict, deist whose God neither reveals nor intervenes is not a 4, but a 4.5).