It’s a very short flight (1 hr 15min — 1 hr 25min) from Britain to Geneva. Plenty of airlines fly on this route, many of them from the small and efficient London City Airport.
Once at Geneva, you are just 100 km from the chalet, via (in approximate order of convenience and expense) hire car, airport transfer, bus, or train.
Make sure you check which side your flight arrives and departs from . Most are from the Swiss side (Secteur Suisse) but occasionally you do get them (notably with Easyjet) from the French side (Secteur Francaise). There is a short walk between the two sides which is not as well signposted as it might be, so it’s best to check — and if you’re hiring a car, the same applies — see Car Hire for more on this.
The terminal is generally very efficient, as you’d expect. It has good facilities including an excellent nursery and extensive duty-free shopping.
http://www.swissair.co.uk — often has good fares, doesn’t charge extra for skis, and has regular flights from London City
British Airways — the Gatwick routes are often very cheap
http://www.easyjet.co.uk — Fly from plenty of regional airports
http://www.jet2.co.uk — Low cost flights from Leeds, now all year round
http://www.traveljungle.co.uk — Good all-round fare search engine
http://www.ski-flights.com/genevaflights.html — Ski flight specialists
