If you prefer not to fly, it’s possible to get the train to Chamonix from London. Although this is a long journey, it’s very well scheduled and runs overnight, so you can go from St Pancras after work, change in the evening in Paris and then go by sleeper.
Alternatively, a new route via Lille to Geneva doesn’t require you to change stations. You can then get an airport transfer from Geneva. Loco2 is much better than Eurostar for booking, I find.
Trains from Geneva Airport to Chamonix
Chamonix is unfortunately not easy to get to from Geneva Airport by rail – car hire, bus or transfer are much easier. However, the rail journey is very beautiful and well worth it if you enjoy railways. There are two possibilities:
Direct from Geneva Airport to Martigny (3 hrs) and then take the Mont-Blanc Express which stops in Les Praz. This takes longer (and is more expensive) but very scenic and has only one change.
Otherwise, a short ride from Geneva Airport station to Geneva main station, and then a tram to Eaux-Vives station in Geneva, where you can get a local express to St Gervais, and then change to the Mont-Blanc Express which stops in Chamonix and Les Praz.
This is not quite a bad as it sounds, as it’s well organised and the railway staff are very helpful, but not ideal with heavy bags. It takes about 2hrs 30mins to 3hrs 30mins and costs very little – about 8 euros one way. The Swiss railways site is much the easiest to use for times etc.