There are virtually no Army service records which have survived from that period. The best you might expect is a record of him being awarded a pension. but since he only appears to have possibly served for a few years, a pension is unlikely. The only way you might find a record of him is to search through the
muster rolls of the dozen or so English Regiments which fought in the War of Independence. However with a name as common as Thomas Allen, this is unlikely to be conclusive, and you wouldn't find any additional details to confirm that you had the right man.
Generally until 1806, soldiers enlisted for life (in practice this meant they remained until they were deemed unfit for further service). A man joining at the age of 18 or 20 might expect to serve until he was in his late 40s.
However the Army's need for recruits during the American War of Independence was so great that new measures were introduced, including larger bounties (£3) and shorter duration engagements. Impressment was also authorised for "
all able-bodied idle, and disorderly persons, who could not upon examination prove themselves to exercise and industrially follow some lawful trade or employment, or to have some substance sufficient for their support and maintenance". However the latter provision was only introduced by the Recruiting Act of 1778, so it probably would not have been the reason for Thomas Allen's absence. That said, even before 1778 recruiting sergeants were not above inducing or tricking men into taking the King's shilling after plying them with drink.
re Rosie's comment, I had assumed that his dates were actually 1753 - 1821.
