Cornish coast for Pam. Heather and gorse catching the sun. H. A. N. W. E. everyone!

OK, here goes.
I have a mate who has an excellent Cockney accent (the working class accent of - originally - the East End of London, as I'm sure you know, Steve).
In about 1983 or 4, he went on holiday to the USA and came back with the comment that although he had had a fabulous time, "they all thought I was Australian; 'What part of Australia are you from?' they would ask me, everywhere I went!"
I realised straight away what the situation was. Just before his visit, The film Crocodile Dundee had made it big in Hollywood. The star, Paul Hogan, has a particularly strong (though perfectly clear) Ausie accent. I would guess this was the first exposure to an Ausie accent for most Americans. They would also not have had much exposure to a Cockney accent. To them, a British English accent was that which was represented by posh private-school educated film actors, or the Liverpool accent of The Beatles (just gusessing on this last point!).
The interesting point is that the Cockney accent of my mate and the Australian accent of Paul Hogan were indestinguishable to them! Whereas to me and all British English speakers they are poles apart!!
The reason is this;
We experienced the Industrial Revolution before anyone else and in the late 18th century there was massive displacement of peoples and much resultant social unrest. Prison populations went through the roof and transportation as a punishment became an easy solution to this problem. When America gained independance, we had nowhere to send our felons. Then Australia was 'discovered', and a collective sigh of relief went up and convict ships duly went forth. A high proportion of the convicts were from the working class people of London with their cockney accent.
200 or so years later and the accents of Cockney London and Australia have each diverged greatly from the accents of those times, so that to me and fellow Brits,they are totally different. Australians too would not understand the confusion!
Hence my mate's astonishment!
To Americans he sounded just like Paul Hogan!!!

Great story, Andy! Yes, it's true that to Americans the accents sound similar. And I'm told that Brits found Dick van Dyke's "Cockney" accent in Disney's _Mary Poppins_ most unconvincing! Van Dyke, of course, was a well-known American actor at the time (ca. early 60s). We also get exposed to British speech patterns from Masterpiece Theater. I've heard that in _Victoria_ and _Downton Abbey_ and presumably others they're careful to distinguish the characters' working class from upper class accents, but that goes right by most Americans! ;)
A couple stories in return: in college one of the grad students was from South Africa, and when I commented his accent sounded British he answered, "To American ears, yes!" And, back in the early 90s, I'd sold some software to a group at the ANU in Canberra. They were having all kinds of problems, and I couldn't get the fax to work, so I finally decided to place an overseas phone call. (Frighteningly easy even back then--no operator required, just punch lots of digits!) Anyway, a receptionist with what to my ears was a very British accent answered, and I tried to speak very slowly and carefully the name of the party I was trying to reach. There was a pause; and then she said, "Oh certainly, sir!" and connected me. Apparently she hadn't been expecting an American accent, and had to back up and reparse what I'd said!

I worked with a few South Africans, Steve and their accent seemed light years away from any English accent!!
Having said that, nearly all North American, South African and Antipodean accents are far easier to understand than many regional accents from the UK! The one I struggle with the most is the Scottish accent from The Gorbals, Glasgow.

Dark clouds and colourful land make a sandwich of sea and distant light sky!