I know this is semi irrelevant as its not in the UK but there are a lot of ring necks parakeets in Tenerife. Although these may be african ring necks that have migrated as opposed to Indian ring necks which are commonly kept as pets in the UK.
Even the black rat has been eradicated from many islands and mainland areas.
causing them to go extinct in the UK barring a single permanent colony in the Outer Hebrides
Well as of yesterday, we have a raccoon dog in the area of Chesterfield, as someone's pet raccoon dog (I kid you not) has got out and gone walkabout!
We no longer have wild raccoon dogs, he's been found. Very battered though poor lad, bloodied and with lost teeth, he was 6 miles from home and appeared to have been run over.
I can't remember where I heard it, but I'm pretty sure I've heard Goshawks prefer Red Squirrels.
I saw a Black Swan in a field near St Brides, Newport yesterday.
Grey squirrel are also slightly bigger, and possibly harder to kill. But I'm sure they'll make use of them if they have no other choice.
As introduced species go, the grey isn't that bad, ecologically. Not great if you're a red squirrel but they fill the same niche and don't seem to harm much else.
Grey Squirrels being bigger makes them slower and a more rewarding meal. It would be interesting to know how they fare against Goshawks in Wales or Northumberland
There are plenty of Goshawks in the New Forest nowadays and I imagine Grey Squirrels must form a significant part of their diet. Being mainly woodland and heathland, Rabbits or game birds are not at particularly high density there. I don't know what else would be a staple prey species for them in this situation.