Phizzy : It is of note to me that they left the quote from a Scholar that said Priestley's group had gone back to "the original manuscripts" without comment.
I think the account was quite balanced. While Dr Malcolm Dick (Senior Lecturer in Local History, University of Birmingham) did say (@ 01:44) "he [Priestley] could compare the King James' version of the Bible with original texts in original languages", that was balanced a few seconds later by the late Dr James Aitken (Professor of Hebrew and Early Jewish Studies, University of Cambridge) who elaborated (@ 01:53) that "there were multiple manuscripts, both of the Hebrew text and of the Greek text, which were now available and showed that sometimes the text that the Authorised Version was based on was not the only reading available".
Priestley's Group certainly had access to Codex Alexandrinus (fifth century), as well as Codex Bezae (fifth/sixth century) and Codex Claromontanus (sixth century). How do we know? Because William Whiston, who was also a unitarian based in Cambridge, published his Primitive New Testament in 1745 based on these three manuscripts. Codex Beza for Gospels and Acts; Codex Alexandrinus for the letters of James, Peter, John, Jude and Revelation; Codex Claromontanus for the epistles of Paul and Hebrews.
If you are interested in Robert Garnham's writings, the only publicly available one I could find was Examination of Mr Harrison's Sermon in a Letter to an Athanasian Christian, 1789, but that isn't about his translation. Maybe more interesting would be the letters between Joseph Priestley and Theophilus Lindsey which can be found here (1769 - 1789) and here (1790 - 1794).