Do you know whether he was officer or other ranks? If officer he'll be easier to trace...If you don't know either way, a quick search of the Army List for say 1813 or 1814 will reveal any officers with that name.
In a quick search in Google Books I've found 2/Lt William Drummond in the Marines, with seniority from 20th March 1797, in the 1800 List. He's still there in 1802 but has risen to 30th in seniority.