I moved to Edmonton last summer, so am basing much of what I write on last year's info, but as far as I know it is still much the same:
The cheapest option may work out as Thomas Cook (Canadian Affair) from Gatwick to Edmonton (Gatwick is the only UK airport that links to Edmonton through the Canadian Affair banner). I used them and it was fine, I paid around £40 extra for Premium, but couldn't overdo the free booze as my new boss was meeting me at the airport!
Air Canada have a daily(?) flight between LHR and Edmonton, although they are often more expensive than Canadian Affair - keep an eye open for seat sales.
Another option is to look out for flights to Calgary - BA and Air Canada fly from London, and Canadian Affair / Thomas Cook fly from LGW, MAN and GLA (GLA just during the summer though?). Once you get to Calgary, its around a 3 hour drive north to Edmonton, or you can catch a bus (Red Arrow is more expensive than Greyhound, but very comfortable), or you can fly with Westjet or Air Canada (about a 30 min flight).
There are more flight option by going via other Canadian cities such as Vancouver and Toronto (they have similar airline options as Calgary), or another option is via Winnipeg (new charter flights via Rekjavik - see Iceland Express website). Note that in these cases you'd then have to book a seperate flight onto Edmonton allowing for suitable connection time etc. off your first flight.
Hope this is of some use!