Contrary to every American Internet debater, Britain is not actually governed by the US Constitution. The applicable law you are looking for is the European Convention on Human Rights, and the 1998 Human Rights Act, specifically Article 10, which starts with the rather bold statement "Everyone has the right to freedom of expression". This is a positive right that the Convention guarantees (subject to the limitations in Article 10 §2) rather than just being a negative limitation on government.
It's a lot more grey than you think. This is very dodgy because most airlines are legally common carriers. The reasons for which they can refuse service are rather more limited than most other businesses. Pubs and bars, for instance, can refuse service on almost any grounds (other than equality legislation: you can't not serve someone because they are, say, black or gay or a woman). But if you hold a valid ticket, have a valid passport/ID and get through security etc., there aren't many grounds on which an airline can decline you from boarding a plane.
Standard IANAL disclaimer applies. The actual law is often rather different from what Internet libertarians believe it ought to be.