Returning to the original question - can't call it an old British usage to say off-ten. I offer W. S. Gilbert's

MAJOR-GENERAL: ...I see where we are getting confused. When you said "orphan", did you mean "orphan", a person who has lost his parents, or "often", frequently?

_Pirates_ opened in London New Year's Eve, 1879, with nary a thought that often could be pronounced with a t; if so, no ambiguity whatsoever, and hence no humor.

Not to say that some people don't pronounce it the other way; just that a British accent isn't the source.

(Or maybe...it was Shakespeare who made us pronounce the T! In Julius Caesar ! "The evil that men do lives after them/The good is often tarred with their bones"?)