The problem is you're going to still have an American accent and you're probably white so you're probably going to have a hard time convincing anyone you're an asylum seeker. If you're brown / black and can pretend you're from Afghanistan you have a chance.

I suppose it also depends a little on what you want to do. In the UK it takes months to process asylum claims but while you wait we'll put you up in a nice hotel, give you food and spending money. So if you just wanted to visit the UK for an extended period of time, it wouldn't be a bad idea.