"He was accused of rape in Sweden.": true.
"They could not prosecute him because he fled the country before he could be charged.": not quite true.
My memory of the events is that JS was first accused of rape, by having two supposed victims talk to the police about this. The police committed a grave mistake (by Swedish standards) when they questioned the supposed victims over the phone: the rules are very clear that rape victims are to be interviewed at the police station and the interview is to be recorded.
For some reason, when the charges against JA were brought up, the prosecutor thought it wise to inform JA by holding a press conference, so that JA found out about the charges by reading/hearing about them in the news.
JA was then taken into police custody (voluntarily or otherwise, I can't remember). He was then questioned, and let go. Between one and ten days later (my memory of the timeline is fading), JAs Swedish lawyer asked the police if he was free to leave the country. The lawyer was told that JA was free to leave the country, since the charges had been withdrawn.
JA then travelled to the UK. By this time, another prosecutor had decided to reopen the case. A very interesting detail here is that this new prosecutor (who has the initials MN) belongs the the same political party as one of JAs supposed victims. When this new prosecutor reopened the case, she wanted JA questioned a second time. JA was willing to be interrogated either by the British police, in the presence of Swedish police, or over internet. Neither option was acceptable to the Swedish prosecutor, for reasons that have either never been made public or that I have forgotten.
JA refused to return to Sweden, citing the possibility he would be extradited to the USA.
I can't see this timeline as JA "fleeing" anything in Sweden: he has given the Swedish authorities the possibility of questioning him, but for some reason they wanted to question him in Sweden, nowhere else would do.
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Apparently questioning a suspect when the suspect is in another country is not uncommon, although the proceedings are complicated by the fact that the "host" country is the one asking the questions, even if the questions are decided upon by the country wanting to question the suspect. The second Swedish prosecutor refused this, for whatever reason she had.