At the moment, a substantial chunk of the London FIR, (Swanwick's airspace), is delegated to EGPX.
Oh, come on! It's not Swanwick's airspace, it's UK airspace (to the extent that it can be called that over international waters). And of that, Swanwick and Scottish are nominated to provide services in only a moderate volume of that airspace.
The original question is an interesting thought exercise that should be done. It's rather like the 'privatisation' of NATS, something that was pushed through by politicians and then left to the industry people to sort out the details. And some of those details are still being sorted out today or are a fudge that everyone can live with.
However, the UK is fortunate in that it has much experience of multiple air navigation service providers working within its airspace, something that has resulted in the UK having a very good regulatory framework for ATS. You may not agree, but try working in an environment where there is a monopoly service provider and regulation is done by the old boy's network and a handshake over coffee/dinner and no-one else knows what was agreed and you might start to value the good bits of the CAA.