Skipping bail is criminal contempt of court in the UK
Penalties are severe:
An immediate custodial sentence is the only appropriate sentence to impose upon a person who interferes with the administration of justice, unless the circumstances are wholly exceptional
He would have been (was?) allowed to argue his case that he shouldn't be extradited to Sweden due to the subsequent risk of extradition to the US. I cannot see how he could possibly convince a court that this would have happened and simultaneously show that the same arguments wouldn't have held water when extradition proceedings were in effect.
I can see he might possibly avoid jail in the UK only by surrendering and going to Sweden. But I wouldn't bet that the UK wouldn't want to extradite him back for contempt of court proceedings (or maybe they wouldn't extradite him to Sweden until after he was convicted and sentenced.