BA has more flights to the US from Heathrow than does any other airline, at any European airport. Heathrow has by far the biggest pax volume to the US of any European airport. BA and AA have 65 percent of that yummy haul right now as it is. They don't want to give any of it up in order to get their massive alliance approved.
Why haven't other airlines--NW/KL, DL/AF, UA/LH--been asked to give up slots in order to get their alliances approved? Because they don't need to. Every USA Cartel carrier that wants can fly to Paris-CDG, Amsterdam-Schiphol, and Frankfurt-Main. There is plenty of competition available to the alliances at those airports. Also, none of those alliances comes anywhere near BA-AA in size.
Only Heathrow, which has the fewest active runways of any major European airport (2) is closed to a limited number of transatlantic carriers (BA, VS, AA, UA). CDG has four runways, Schiphol 5, and Frankfurt 3. The alliances at these airports can be held accountable by competitors if they tried to gouge passengers.
Fortunately, the US Department of Transportation, which finally grew balls and stopped being the floosie of major airlines, told BA and AA they'd have to give up slots at LHR if they wanted approval.
Unless *every* USA Cartel carrier can get access to Heathrow, like they have access to CDG, SPL, and FRA, *NO* BA/ AA! For now, there won't be. But BA and AA will probably try again sometime.
Need a new airline paint scheme? Better call Saul! (Bass that is)