Between leaving Sweden and entering the Ecuadorian embassy Assange spent several weeks in UK, while a British court decided on the validity of the European arrest warrent against him. Plenty of time for the US to demand his extradition.
Not weeks, years. He left Sweden in 27 September 2010, and it wasn't until 14 June 2012 that he had exhausted his rights of appeal in the UK (short of any appeal to the European Court of Human Rights in Strasbourg). He was on bail for pretty much the entire period (he spend nine days in police custody prior to being granted bail, but I don't believe he was ever in custody after that).
(Disclaimer: I don't know nothing, I just think I heard.)
If he were to be extradited to Sweden from the UK, and if the US wants him extradited from Sweden, the UK would have to give their permission to do so. Sweden would simply be an extra complication and wouldn't help any extradition claim the US might have.
http://jtl.columbia.edu/wp-content/uploads/sites/4/2015/04/F...
Which place he gets sent comes down to the decision of someone, likely the Home Secretary, but my memory of this is fuzzy and it could be down to the judiciary.