Depends on how you define Britain's as in UK registered, UK Owned, Operating in UK with lots of flights, UK AOC etc
I know they are the largest in passengers carried by a UK registered airline (factor I am questioning as per link) but also in the number of passengers carried in and out and around the UK.
By that I could mean any airline. e.g.) KLM will carry more passengers than Manx2, Lufthansa more than Eastern etc.
I am not bothered from which country the airlines come from.