After receiving very useful help from this forum on this topic, I forgot to report back with my findings. A key resource was Encyclopedia of British Provincial and German Army units 1775-1783' (Katcher) referred to by jf42. My ancestor was German and I thought that he might have transferred into the 57th regiment of foot from a Hessian unit, but that seemed a little unlikely for the British army to do right at the end of the war. Then I came across a section on the "Scheither recruits" in Katcher.
So here is the info that I found:
In the autumn of 1775, the British government contracted Georg von Scheither, a Hanoverian colonel with experience in the Seven Years War, to raise 4,000 men from across Germany to help bring British regiments up to full strength. Scheither’s recruits, which in the end were fewer than 2,000 men, were sent in small groups to Dover to be reviewed, and then shipped to the colonies for distribution into the various British regiments.
The embarkation list in May 1776 shows my ancestor, Georg Richter, aged 18 years and born in Saxony. He had previous military service in the Electorate of Saxony in the Holy Roman Empire (“Liste Des Recrués Anglois embarqués à Stade pour Spithead en Irlande ce 14me de Mai 1776”, War Office (WO), Class 43, vol. 405).
Georg Richter and the other Scheither recruits arrived at New York on 20 October 1776, and he was enlisted as a private in the 38th Regiment of Foot on 21 October 1776 (WO, Class 12, vol. 5171). Each British regiment received about 40 of the Scheither recruits, which were usually distributed evenly across the ten companies of the regiment.
Richter spent most of the war garrisoned in New York. He may have participated in some battles (still to be verified). The guards and light infantry companies of the various regiments were often detached to form battalions that saw much more action, but Richter was in a regular company.
On 25 August 1783, Richter transferred from the 38th to the 57th Regiment of Foot, still as a private. The muster roll for the 38th Regiment of Foot shows that he was discharged on 7 September 1783. There is a two-week period where he is listed on muster rolls for both the 38th and 57th. It was common to enter a later date than the actual discharge date in order to pay the discharged soldier extra “to carry them home.” He was entered into the muster roll of the 57th as George Rector and used this anglicized form of his name for the remainder of his service (WO, Class 12, vol. 5172 and WO, Class 12, vol. 6633).
The 57th Regiment of Foot moved from New York to Nova Scotia in September or October 1783. George Rector remained in the regiment until he was discharged on 5 July 1786 (WO, Class 12, vol. 6634). Since the 38th Regiment was returning to Britain, some soldiers were probably given the option to transfer into a regiment that would be posted in Canada. After his discharge, he applied for and received a land grant.
Although the Katcher book first clued me in to the Scheither recruits, the article "Vcn Scheither Recruits in the 38th Regiment of Foot" by Don N. Hagist (Journal of the Johannes Schwalm Historical Association, Vol. 13, 2010) was the clincher.
I want to offer a much belated thank you for the help.