I’m no international law expert, but wouldn’t the UK demand that the US disclose in advance everything they plan to charge him with?
Otherwise, couldn’t he argue in the UK that his rights might be violated by the extradition, in the event that the US charges him with something that isn’t considered a crime in the UK?
Assange, and his lawyers, made all kinds of arguments and appealed his extradition to Sweden all the way to the High Court and lost. He hasn't actually had any court time related to the U.S. extradition because he fled. I would assume that if he had a decent chance of winning on the merits for the U.S. extradition, he wouldn't have spent 7 years hiding in a cupboard.
Otherwise, couldn’t he argue in the UK that his rights might be violated by the extradition, in the event that the US charges him with something that isn’t considered a crime in the UK?