Apparently he's asked for the trial to be held here, back in 2010, and as he's not committed a crime here, the best UK can do is try to extradite him. Which he went to court for many times, and UK were trying to find an alternate solution to sending him there, but Sweden want him there.
I've read a bit more the last couple of days since it's been in the news a lot about the Swedish women, and give the 'facts' (what's in the news) I'd say he should go there and give his side of the story, or even just video-link it maybe? We do it for just a few miles, and it's the same situation if you needed to do it for 1 billion miles, you'd still be able to give him a trial. And that's what makes me think they do perhaps want to extradite him.
Though I can't find the articles right now, I did read a few that had quoted in 2010 that one of the 'victims' was seen with a US official, who at the time would not have been on the case unless it was connected to Wikileaks and the leaked US documents. I'm trying to find it now but it's getting late to be fair so struggling to dig deep. That's what I've always remembered with regards to this situation and what made me think US want him sent to there but UK are being a bit too pedantic with old laws etc.
Clearly if he's raped a lass, then he needs to get sent down. But in sweden not America.
Though having said that, perhaps he is guilty and he's avoiding going to sweden because if he does end up in prison for years, its all time the US can spend on putting a case together to extradite him. So perhaps there's that side of the story too maybe.