NickB

That depends on whether the UK already has a live agreement in place or not. If it has no agreement (as, for instance, in the case of the US), it has to conclude a new agreement. It can negotiate with the US whatever it wants/whatever it can in terms of who will be entitled to fly under the agreement. It does not have to limit it to UK-owned airlines and can negotiate a different kind of clause.
With respect to countries with which the UK currently has an agreement, many will already have a clause allowing EU-controlled airlines to have traffic rights between the UK and the third country. These agreements will have to be amended to change the ownership and control clause to replace the EU ownership and control clause with a UK ownership and control clause. There too, nothing obliges the UK to restrict traffic rights to UK-only airlines if the other country is OK with a wider clause.
Perhaps the simplest for the UK would be to negotiate clauses that include UK-owned and control airlines and add specifically BA in the clause or allow the UK to designate an EU-owned and controlled airline in addition to UK-owned and controlled airlines. This would be one way to solve the problem that BA is controlled by IAG (assuming IAG is controlled by EU nationals as this will be necessary for IB to retain traffic rights within and to/from the EU).