Here's the kicker - the UK's permission would be needed if Sweden went on to consider extraditing to the US http://www.bbc.co.uk/news/world-europe-19426382
It's because we're still vicariously liable because we're handing him into the power of another country.
So not only would Swedish courts be an appeal process the UK's would as well.
Plus it is worth making the point that it is inconceivable that the USA would "black bag" Assange. Kidnapping a high profile Australian citizen from a European country while he was still being protected by the UK legal system is beyond international incident territory, the repercussions would be immense. If the US wanted to illegally abduct him making him the focus of the international media and several legal systems is a really bad precursor.
If anything they want to put him on trial and make an example out of him, not apply car batteries to him in some Egyptian hell hole. A trial means a legal means of getting him before a US court, which means following established extradition processes.
Apart from academically, it is largely irrelevant whether or not this is likely to happen.
What is relevant is that there are good reasons why Assange might genuinely believe it likely to happen, which means there is good reason that his asylum in the Ecuadorian embassy was requested in good faith rather than "just" to avoid a Swedish court case with a low risk of conviction and short sentences were he to be convicted (he'd have been out a long time ago if he had been extradited and convicted in Sweden).
So not only would Swedish courts be an appeal process the UK's would as well.
Plus it is worth making the point that it is inconceivable that the USA would "black bag" Assange. Kidnapping a high profile Australian citizen from a European country while he was still being protected by the UK legal system is beyond international incident territory, the repercussions would be immense. If the US wanted to illegally abduct him making him the focus of the international media and several legal systems is a really bad precursor.
If anything they want to put him on trial and make an example out of him, not apply car batteries to him in some Egyptian hell hole. A trial means a legal means of getting him before a US court, which means following established extradition processes.