This kind of information changes very frequently, depending on how the owners of EDI price FastTRACK access and what airlines are willing to pay or pass through to passengers. Even at Heathrow, a vastly busier airport, access may change in a matter of months.
FlyerTalk is generally reliable for such information, and according to various threads as of June 2013, FastTrack access is reportedly complimentary for
- British Airways Silver and Gold members, and those ticketed in Business UK, Club Europe/World, and First Class (I believe BMI Diamond Club has been fully integrated into the BA program)
- Sapphire and Emerald-level members of OneWorld airline programs, anecdotally— EDI is not listed on http://www.oneworld.com/fast-track
- EasyJet passengers with "Speedy Boarder" access or EasyJet+ cards
- Flybe passengers with "Flybe Plus" cards
- Air France Gold and Platinum members flying on CityJet
You can buy Fast Track access from the vending machine for £5, and it is also included with some parking packages.
I could not confirm any information for the other major airlines there, which include Aer Lingus, Air Canada, Germanwings, KLM, Lufthansa, Norwegian, Ryanair, Scandinavian, Thomas Cook, Turkish, and United.