The English battle to get rid of this invasive species : The Epic Century-Long English Battle to Rid Itself of American Squirrels | Atlas Obscura
Not quite correct reference to the Isle of Wight here-historically there have never been any Grey squirrels present there so they have never had to eradicate them. About twelve years or so ago there was rumour of individuals being seen in the West Wight, but despite hair test traps being set up, their presence was never confirmed then or later. But equally importantly a single (gravid) female was supposedly found dead on one of the roads-though its presence was never explained or publicly confirmed.
Honestly I feel like we deal wit the greys in a very half-hearted manner overall. There's so much public divide and politics attached to the whole affair. There's also a big country/urban divide as well which never helps as the urban population nearly always out-votes the countryside. I know countryfile did a "BBQ squirrel" show a while back and there's loads of noise about how pine-martins might be the solution (greys can't run as far to the end of tree branches as reds can and martins can thus get at the greys and not so much the reds); however the pine-martin has suffered at the hands of our gamekeepers/farmers for many years and it will take a big turn around with a lot of support from them to get them back. That said we've got a lot of birds of prey back that we never thought would be possible so pine-martins have a chance.
Pine Martens are supposed to be extending their range quite rapidly nowadays. I think they could represent the only effective solution to the Grey squirrel problem with their 'natural selection' method of preying on squirrels- they could just succeed where all the other efforts and methods have failed, but they would need to be widespread over the whole UK to have a chance of success.
Aye their range can increase rapidly due to nothing filling their niche; but unless countryside workers accept them they'll be beaten back again like they were before. At first it might not be too hard; but there's huge money in pheasant shoots now; big money that already poses problems for raptors. It's also a showing that, to my mind, we have allowed the grey to spread as it has. Like most things if enough focus is brought toward a species directly we can remove even high population ones from an environment ( one can say we are pretty darn good at it actually).