I think us Americans say "There was one in particular who must have been getting on in his years because he'd seemed pretty ancient when I was there." instead of just "getting on". Technically, "getting on" is also American English according to several American English dictionaries I have, but in reality, I never hear any other Americans say it without following it with "in years" or "in his years", etc. Or at least, I never really hear it in my part of America as just "getting on". It might appear more often in the short form in British English though as atarilynx explained. :)