We have a guide on my site Loco2 which might also be useful: http://loco2.com/blog/2012/10/ski-trains-london-to-france-alps
Also, I recommend checking out http://www.snowcarbon.co.uk which is also packed full of information.
Yes, it can be done quite easily! Seat61 has a great guide to doing so, which is well worth a read!
On weekends during the winter, there are two direct Eurostar services between London and Moutiers, Aime & Bourg St Maurice. There are daytime trains both ways on Saturdays, or an overnight one out on Friday nights, back on Saturday night (but it doesn’t have beds, just seats). A large number of the popular resorts are just a short transfer away from these stations (tens of minutes, not several hours!)
Another option is to take the Eurostar to Paris, then either a daytime TGV to the alps, or an overnight sleeper (with couchette beds!). This does involve a cross-Paris transfer though, which might be an issue depending on luggage – it’s a single metro ride with a few flights of stairs to cross Paris. Alternately, for the Swiss alps, and some bits of France (eg Chamonix), you can take a Eurostar then Lyria TGV, with a cross-platform change in Lille.
These services are generally unaffected by the weather – in January 2013 when there was a lot of snow in the UK and northern France, lots of flights were cancelled, but the daytime Eurostar was only a few minutes late into Bourg!
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