BA is a Spanish airline, headquarters in Spain and run by a Spaniard. It is a relevant factor in whether the economic benefits actually pass through to the British people who will pay for the airport expansion and who will suffer the environmental consequences of the airport expansion.
As far as I am aware, BA is registered in the UK and headquartered in the UK (and run by a Spaniard). BA is however a subsidiary of IAG which is headquartered in Spain and run by an Irishman.
Heathrow airport is a holding company with no UK ownership at all (as far as I can see). In addition, it is likely that any new runway would be funded by them (at least principally).
Spot on. BA is a UK registered company operating flights under a UK AOC, and run from offices at Waterside, Harmondsworth, near Heathrow.
It is a wholly owned subsidiary of IAG, a company whose registered office is in Madrid but whose operational headquarters are in fact at, wait for it, Waterside, Harmondsworth, near Heathrow. IAG is quoted on the London and Madrid stock exchanges, the largest single shareholder is Qatar Airways with 20% of the shares. Yes, the CEO of IAG is Irish, the same man who has in the past run BA as a standalone airline, and before that Aer Lingus (now another IAG airline). Hard to think of a more genuinely multi-national airline group, really.
Apart from the Irish Willie Walsh, standalone BA also had in its past the Australian Rod Eddington as CEO. I really fail to see what the nationality of the CEO has to do with the "nationality" of an airline; if having a British citizen at the helm is the qualification for an airline to be British, then welcome to Sir Tim Clark's Emirates Airlines as Britain's flag carrier.