My non-native ears can't figure it out, either To me he really sounds "American", there seems to be some uncertainty the way he speaks that would have made me think he's not a native. But he does a good job anyway, for it's really difficult to imitate accents, even in my own mother tongue I'm able to imitate convicingly not more than 2 or 3 accents.
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I guess non-British people would hear something a bit off with the cockney accent but not realise how awful that is either...What exactly is there that makes his Cockney sound awful? I think it would be important to focus on where he actually fails in his attempt to mimic different accents.