If I were in your situation, I would confront him about it, not so much her. While some of the frustrations seem to be aimed at her for trying to initiate and maintain a disproportionate amount of contact with your fiance, it seems that the big issue is that your fiance is going along with it and not putting his foot down...or only making a half hearted attempt to put his foot down only after you approach him about it.
Personally, I don't understand why it's so difficult for him to cut off (or reduce) contact with someone who is:
(1) not even on the same land mass,
(2) on internet sites that have blocking features, and most importantly,
(3) someone he claims he doesn't even like talking to.
That, IMO, is worrisome. If he claims he doesn't even like talking to her, why does he chat with her daily? Why does he type a mile a minute on Facebook chat with her?
Is he afraid of confrontation? More afraid of confronting her than you about this? It appears she broke up with him (for her boss) - is he not completely over her yet?
Maybe it's just me, but I can't reconcile him claiming that he doesn't like talking to her, and then him furiously having a Facebook chat with her, if that's what he's doing.
Even with the visa paid for and the plane ticket booked, if I were in your situation, I would resolve this ASAP, even if that meant delaying your move to the UK and taking a financial hit on the plane tickets and visa. While it may seem to be a problem now, if you moved over there and it didn't work out, it would be a bigger problem then.
I would tell him (again, if I were in your shoes) that he should commit to the person he's engaged to, and make good on it. And if he can't or won't do it, it would be better to find out now.
IMO, he's not treating you like his fiancee, and he's not showing you respect or taking your feelings into consideration - how upset you are by his actions. He may truly never go back to her physically, but it just seems to me that he's investing a lot of emotional energy into interacting with her, and because their interactions are straining your relationship together, his behavior is not acceptable if he wishes to maintain a healthy relationship with you. There's a difference between remaining friends with an ex, and what he's doing - to the point where his current partner feels upset.
EDIT:
Glad to hear you've sorted it out some.
Do you believe her?