To sue them you can file a County Court claim online at
https://www.moneyclaim.gov.uk/web/mcol/welcome
Money Claim Online require that all reasonable attempts to resolve such a matter out of court are exhausted before filing, when I successfully sued Eon previously I wrote to them asking that they confirm that all reasonable attempts to resolve this outside of County Court have been exhausted whilst letting them know that "I'm taking you to County Court you baskets" ... or words to that effect.
As soon as you file and pay the online fees and the subsequent court fee, and on the proviso that BA are going to lose, then they're going to be paying your court and other legal costs, travelling expenses, their lawyer fees etc.
To quote my ex boss Sir Freddie Laker, when referring to BA, "Sue the Bastards"
