I can contribute info on the leg from Paris to Chamonix
There is a train that leaves Paris on Saturday evenings in the summer at about 11pm and drops you off at St Gervais Les Bains, a short train ride from Chamonix, at 6am the next morning. It’s only £30 each way if you book well in advance and don’t mind sleeping in a chair (that reclines a long way). It’s £47 for a couchette, as I recall.
The return train departs St Gervais Les Bains at about 8.30pm on Saturday and gets you into Paris at about 6am the following morning.
The Saturday evening trains are by far the cheapest way to do this, the trip from Paris is £60 return plus maybe a bike fee. If you go at any other time it’ll be north of £150.
I’ve no idea how the price compares to options from the UK, but from Belgium where I live it’s the most economical way of doing it if you’re solo, or the rest of your party is coming from the UK etc. It’s a little cheaper than flying, you get an extra day and there is less hassle. And I like trains.