It’s a general principle of extradition law that when a country surrenders someone to another country, it’s only for the purpose of the crime alleged in the extradition request. The requesting country can’t “bait and switch” - ask for extradition on a relatively minor matter, then once the person is extradited, charge with a much more serious crime, or a greater number of crimes. This principle is called the “rule of specialty.”
One aspect of the rule of specialty is that the requesting country cannot then turn around and extradite him to a third country. The US recognizes this principle, as this document from the State Department[pdf] states:
Note: I’m not sure what this document is; appears to be an extract from a US extradition treaty.
Britain’s Extradition Act 2003 sets out this principle as well. It divides countries into 2 categories. Category 1 countries are the European countries that have agreed to the European Arrest Warrant system, and includes Sweden. Category 2 countries are other countries that Britain has an extradition treaty with, such as the US.
Section 58 of the UK Act sets out the process that must be followed if a Category 1 country such as Sweden has received a fugitive from Britain, and then applies to Britain to extradite that person to a Category 2 country, such as the US. The Secretary of State essentially has to decide if the accused could have been extradited to the Category 2 country if the accused were still in the UK.
So, given all that, what’s the point in some secret conspiracy by the US to try to get Sweden to extradite, and then apply to Sweden to extradite?