We were always told, when I lived in Fulham, that a real hard core Fulham accent was quite different from E. London. I have never tested this, but I can vouch for the fact that the Fulham one is amazing - I hear builders using it, and I think they are putting it on. Maybe, maybe not. Now your Putney accent, quite different, of course.
I also find myself echoing their accent, bit embarrassing really, but I think I do it, wherever I am.