ATC are a fairly vital part of the safety chain, so right now another company (NATS) in another part of the world is a large part of the control of many overseas airlines assets.
Depends what you mean ultimately by control. At the moment the airline has two of their own employees on the Flight Deck of said asset and they in turn have the option of accepting any revised clearances ATC care to issue..or not ( even if the mighty data link is used by ATC to pass a reroute up to the FMC the "execute" button still needs a press..ooooh, the power!!!!!
). Given ATC's peformance in some parts of the world
I rather suspect the airlines would be
very reluctant to lose that power of veto.
I.e. the ground-based pilots tasked with intervening in an emergency that the automatics can't handle would work for and alongside ATC for that airspace.
IMHO that would rely on 100% comms reliability and if you move away from "flight following", with the "pilot" in company HQ simply available in case of a hopefully rare emergency or the need to liase with ATC, and move more into the realms of having the "pilot" in company HQ monitoring every clearance I suspect it would simply be cheaper to put the pilot, or even pilots, back on the aircraft.