In reply to Petarghh: Probably the easiest way is London to Paris on Eurostar, then Paris to Geneva on TGV, short hop to Geneva airport and get a pre-booked transfer from there. Getting the train all the way to Chamonix is possible, but the last bit is very slow. Or you can get a sleeper from Paris to Chamonix, that's a good option, often cheaper, and doesn't matter that it takes ages as you're sleeping (take earplugs and eye mask), gets to Chamonix about 10am, leave Paris Austerlitz about 10pm.
I've taken ice axes on eurostar many times without any problems, although admittedly not for 3 years or so, so it might be different now.
I book using eurostar.com and voyages.sncf.com
French tickets are released 3 months in advance, and eurostar 4 months, I think.
Cheers, Es.