I once had the opposite experience. In the impressive entrance lobby of the hotel that I was staying in on a business trip in Malaysia, I noticed a discreet sign pointing to 'REST ROOMS'. I had, at this point, never been to America. Being inquisitive, I thought I would explore. Oh, it's a toilet. Not a rest room. Not sure what I had been expecting to find, but definitely not a toilet.
Malaysian English also takes some getting used to, and now has its own name - Manglish. Most Chinese Malaysians speak Malay, English (apparently), and Hokkien (so you should now know that I was in Penang). I've had the experience of walking along with a couple local colleagues, and not really paying attention to what they were saying. After all, being English, I only speak English (and a very limited amount of French which is not much help in Penang). Then being aware that they were waiting for an answer from me. Oh - that was English, was it? Not Chinese or Malaysian!
And other English colleagues who visited have had similar experiences.
Getting back to the OP:
The British generally understand Americans, or at least, we think we do, because there is much more American TV in the UK than vice visa. Our problem is not with the American accent but with words having different meanings, although often related. And we probably miss many American cultural allusions.
Perhaps a more interesting question is: "do British people ever have problems understanding each other?" (when both speaking English). Definitely. Although the differences in accents (and dialects) are becoming less. Not sure how easily someone from Yorkshire would understand someone speaking in a broad Devon dialect.
But dialect is more than just a difference in accent or phrases.
