There are quite a few people who are wrong on the issues here
The rape claim against him would definitely be a crime in England as well, if her story is true. He stayed with one of the women, and one night they had consensual sex with a condom. The next morning she woke up to find him inside her again, this time without a condom. As she had made clear to him before they'd only have sex if he wore a condom. Remember that English law requires consent for intercourse not to be rape, whereas Swedish law only makes it rape if consent is denied. So "this wouldn't be a crime in the UK" is bullshit, and, as stated above, rape is also one of the crimes where such an issue is not taken into account for extradition.
The women are not pressing charges but criminal cases like rape don't require them to do so, as it falls under public prosecution.