The Swedish investigation may well expire (parts have already). This has never really been about the Swedish charges.
The English charges (relating to failing to comply with bail) do not expire. Very few offences in English law expire - something I'm sure will surprise Americans who are used to relatively short statute of limitations on even some serious crimes (e.g. the Bill Cosby sexual assault allegations). England routinely prosecutes and jails people for alleged offences 50 & 60 years ago; many are elderly men in their 80's & 90's with little chance of mounting any sort of defence due to their age, but that's another issue.
Arrange cannot wait out the English charges and England is well documented as an enthusiastic participant in torture and rendition of both it's own citizens and foreign nationals. The UK-US extradition treaty is a shocking travesty of justice and is routinely used to send people from the UK to the US for activity carried out in the UK and considered lawful in the UK but which has upset US authorities. [1]
He has good reason to remain in the Embassy. Unfortunately he's likely to be there for many years to come and either be grabbed when he requires hospital treatment at some point in the future, or, the UK will decide it can live with some diplomatic protests and expulsion of UK diplomats after sending the SAS in to grab him from the embassy and hand him over to the US.
[1] http://www.bbc.com/news/uk-scotland-tayside-central-14339901