The interesting thing is that Sweden could argue that they get first bite at him.
The law gets a bit murky on this because there is a single count of rape that they can charge him. Does that take priority over the US? Maybe. So then what happens? Even if the US loses in the UK (doubtful) they will get additional bites at him in Australia where he would have less of a chance to win an appeal.
And this isn't bending over for Trump.
The indictment against him was during Obama.