Just to play devil's advocate here: How are Sweden supposed to promise not to extradite him to the US? Wouldn't that depend on what kind of possible extradition request would arrive, if any?
I know nothing about Swedish law but, as an EU member and ECHR [1] signatory, Sweden cannot extradite a person to the US (or any other country) if a possible punishment that they could receive would deprive that person of their rights under the convention. In practice, this applies to judicial execution, detention without trial, or detention without the possibility of release. It may be that the US could give underta…
http://europa.eu/legislation_summaries/justice_freedom_secur...
Why someone would flee Sweden to go to the UK when they are afraid of US extradition is beyond me. The UK has a much closer relationship with the US.