I'm always minutely perplexed when I meet someone who likes to fake accents as an ice breaker. It's not limited to British accents, of course, but British accents seem to be the most popular choice. I suppose the Eastern European accents might be harder to maintain and the Irish brogue is just too hard to nail.
More fascinating is that I've never seen anyone call someone out for it. Everyone sort of pretends it's not happening.