The “the rape allegations were made up so Sweden can extradite me to the U.S.” line never made sense. The UK is even friendlier with the US than Sweden is, there's no way he couldn't be extradited from the UK.
It is also worth pointing out that the UK has to give permission for onwards extradition from Sweden to the US if a request were to be made. If the ultimate game plan for the US was to engineer his extradition it seems a little odd to not have requested it from the UK alone while he was here, rather than have to throw the Swedish legal system into the mix as well. Both Sweden and the UK are required to not extradite when there is a risk of the death penalty being applied - so that wasn't an issue either.
The whole "but the Swedes won't guarantee not to extradite him" is neatly covered by the whole point the US hasn't requested an extradition, and no legal system can or will get involved with hypothetical situations.
As I understand it, the UK government has to give permission for extradition from Sweden, but extradition from the UK to the US requires the support of both the UK government and the UK courts - and extradition requests from the US are held to a higher standard of evidence than ones from the EU. The UK government is certainly pretty friendly to the US so that's no obstacle, but the UK courts are less predictable.
The whole "but the Swedes won't guarantee not to extradite him" is neatly covered by the whole point the US hasn't requested an extradition, and no legal system can or will get involved with hypothetical situations.