I'm not sure if the red squirrels in the UK are the same species as in North America.
Around here, they are sometimes called fox squirrels. The gray squirrels have seemed to mostly settle in urban areas, and are the typical squirrel seen running along telephone wires, being fed, digging millions of holes in lawns etc, in towns and parks.
Red squirrels are mainly found in more outlying "wild" woods.
So they seem to each have a niche, once they get it sorted out.
This maybe the key to the survival of the grey since towns and cities have small wooded areas and gardens for them to thrive but the habitat of the red is dwindling and they do not thrive in towns and cities wooded areas.
Seems the species might be equivalent then.
I just don't know if the reds don't thrive in towns because the habitat itself is unsuitable...or if they don't thrive there because the grays find it more suitable, and thus aggressively take it over.