The elephant in the room is BA and trans-atlantic traffic rights. Divesting enough shares off IAG to make it 51% EU owned and controlled is a big task as it is, made no less complicated by having a single non-EU/non-UK shareholder owning than 20% of IAG (Qatar Airways). However, finding a way to make BA 51% UK owned, whilst simultaneously finding a way to have Iberia, Level, Aer Lingus and Vueling 51% EU owned and controlled, is an impossible task. Put simply, IAG can't be EU and UK controlled at the same time. One could come up with a construction where the individual airlines retain separate ownership, but that would also spell the end for the IAG umbrella.
If BA is less than 51% UK owned and controlled post brexit, transatlantic traffic rights to the US is in serious jeopardy. If IAG is less than 51% EU owned and controlled, Vueling, Iberia, Level and Aer Lingus cannot operate intra-EU flights, which represents 90% of their combined operations.
One possible solution could be to take BA out of IAG completely, with all the competitive disadvantages that would bring. At this point in time, that seems like one of a very few possible scenarios.
What this will undoubtedly mean is, that if you're an airline employee working for BA, your future is in serious jeopardy. Then again, the same can be said for almost every UK based employee and business owner.