That won't keep him out of Sweden. And it's becoming increasingly clear that all of this is about avoiding facing those rape allegations in the Swedish courts. And he's already lost that case - so long as Sweden re-issue that EAW, he's off to sunny Sweden with no right to appeal.
I think he'd have a pretty good chance of fighting extradition to the US in Sweden - I'm not sure the UK governement would lift a finger to help him. So long as the US promise not to seek the death penalty. But he'd be able to tie it up in our court system for many years - and of course would have to do that from prison - as there's no way he's getting bail again.
But I've no idea if the US will bother. They'd be stupid to, unless they've got cast-iron proof that he was actively assisting Manning in hacking the army computer system. And it had better be good, given the way Manning was treated - as they've got to establish the credibility of her testimony to standards acceptable to a civillian court.