I'm British and I think your British friends are wrong. I do agree that most of the people who use "orientated" are British but I don't understand where they get that from.

"Orientation" is the term for the process of orienting something. So a design process in which, say, user stories are turned into object-oriented designs could be called "object-orientation". But it does not follow that the result is "object-orientated" - we'd just call it "object-oriented", in the same way that something which has been through a transformation is not "transformated" but "transformed".