In reply to Daniel Iveson:
Fwiw I did a five day intro to summer alpine mountaineering course and I was the only one who currently climbed or led trad, and only half of us had done winter skills before. Personally I think those were useful for me (I'm a slow learner and it might have been a bit much otherwise) but the guide generally seemed more impressed with those who were fit, good with exposure and picked things up quickly.
I won't suggest that particular course as I don't think it really left me feeling like I could do easy alpine stuff independently afterwards - I think it may have been partly due to weather, but we didn't do as much glacier/snow stuff as I would have liked. But just wanted to reiterate that your starting point sounds like a good one.
Roped scrambling courses so you know how to do alpine coils and short rope etc would be a good skill to learn in the UK (not something I had, as I only ever did trad, or scrambled grade 1s).
I'd also add the perhaps obvious point about consolidation - if you do winter skills etc (not a bad idea), make sure you then use it - not like me, doing it in January 2020, and then doing a mountainteering course before a major injury!
Post edited at 15:49