It could go either way. There is no set rule for this. Were I to guess, I'd say that the UK will find it in their best interest to extradite him to the US rather than to Sweden. That's simply because of the UK's and US's "special relationship" and because the US is in a position to put a lot more diplomatic pressure on the UK than Sweden is. That doesn't mean that the crime he is alleged to have committed in Sweden is a lesser crime.
What will probably happen is Assange will first face his charges of espionage in the US. Sweden, which just reopened its case in May, will continue with its investigation and continue gathering evidence while that happens. Whether or not Assange is found guilty of the charges of espionage in the US, if Sweden still wants to question Assange, which is the current situation, the US will then probably invite Sweden to question Assange in the US, but if Sweden wants to charge Assange with rape, which Sweden hasn't done and isn't at the point of doing as of now, the US would then likely extradite Assange to Sweden to face those criminal charges on the condition that, if he's been convicted of espionage in the US, he be returned to the US to serve out his prison sentence in the US, after the completion of which he'll be extradited back to Sweden to serve out any prison sentence there.