Says who? Sweden? The UK? Assange? The US?

UK is in Five Eyes, Sweden is not. The UK is a co-founding member of NATO, Sweden is not in NATO. The USA & UK have a "special relationship", Sweden & USA don't. UK helped the USA invade Iraq, Sweden did not.

Half of which if not all of which had absolutely no bearing on whether or not Sweden were more or less likely to extradite Assange as a result of pressure to do so by the US.

The "special relationship" is not a cover-all term for the US & UK to do anything on a promise like a teen couple in love. It covers a lot of military and trade stuff, which wikileaks is a part of, but in this case Sweden clearly acted as a player in a larger game. Assange seems to have presumed an ulterior motive in the charges brought against him and proceeded accordingly.

People have claimed that "Sweden has links to NATO" as a reason why Sweden might extradite Assange to USA[1]. So, yes, the fact that the UK co-founded NATO is very definitly relevant.

[1] https://news.ycombinator.com/item?id=7988782

Is Sweden a member of "Five Eyes" ?

No, it is made up of anglophone countries (Canada, UK, US, New Zealand, Australia)

But is still a close ally of the US when it comes to intelligence.