The word falcon, when it first appeared in Middle English, was spelt and pronounced l-lessly: faucon, faukun, faucoun. The l was added in the 15th century to accord with Latin, and presumably the pronunciation changed (for some speakers at least) subsequently. Faucon is still the word in modern French, and is pronounced /fokɔ̃/.
The OED offers two pronunciations, the traditional RP pronunciations: /ˈfɔːlkən/ (roughly "fawlken" for those who haven't bothered to learn the IPA) and /ˈfɔːk(ə)n/ ("fawken").
However, the OED's entry hasn't been "fully" updated since it was written in the 19th century, and more recent British dictionaries offer different pronunciations:
American dictionaries give these pronunciations:
- Merriam-Webster and American Heritage both offer non-IPA transcriptions equivalent to /ˈfælkən/, /ˈfɔ(:)lkən/, and /ˈfɔ(:)kən/
- Cambridge (AmE version) has /ˈfælkən, ˈfɔl-/ ("falken" with a short "a", and "fawlken")
It seems a range of pronunciations are possible in both countries, although all American sources put /ˈfæl·kən/ first (which is sometimes an indicator that it is the most common or most widely accepted option, although this isn't infallible), while British sources vary.
Even though all the pronunciations are used and heard, this doesn't mean all will be equally accepted by everyone. At least in Britain, it's not entirely unknown for people to make fun of (or make secret judgements about) others' pronunciations if they are unfamiliar with them or if they think they sound too "posh" or too "common" or too northern or too southern.