There is no 'choice' available
That's true for flights starting and/or finishing in the UK - they have to use NATS for at least part of their route.
Overflights have a choice.
Operators take many things into account when deciding the routes to fly, including the ANS costs, the fuel burn and the ANS quality of service. When fuel costs are low, the UK mileage is minimised. When fuel costs are high, the mileage flown is minimised. When French ATC is on strike, more will bypass France by flying oceanic routes or going via Belgium and Germany.