If this is the street you are referring to (Google Street View & OSM history), it looks like it recently was marked as private access (maybe by yourself?) and thus the next time Portsmouth pulls in new data (~every 5-6 weeks) it will be removed from CS list.
In my opinion (from 6203km away) I think this street is “legal” to include for CS purposes and I wouldn’t mark it as private access in OSM. Usually when I think of a ‘private’ access street, I think of streets where if I ran down it, I would get yelled at by somebody. On this one you may get weird looks from someone on the loading docks but likely no security guard would chase you down.
The street looks to be a part of the Queen Alexandra Hospital campus, which is apart of the National Health Service, which is a publicly funded system (going based off Wikipedia pages). I doubt the hospital itself is charged with maintaining the street (re-paving, painting lines, etc) but likely relies on the local government to do so. In addition the street has signage with its name on it and I don’t see any signs in Google Street View that specifically say no or limited access for pedestrians or other forms of transportation; like you may see on a street for ambulances only. Lastly there is a sidewalk for half of the street that looks to lead to certain entrances of the hospital so foot access looks to be ok.
If I were marking the street, at most I’d set the access to ‘permissive’ (see here for description of access values). This would still lead to it showing in CS but I think that would be the proper marking value in OSM. I’d recommend checking out other hospitals in the area to see if other OSM users have marked similar roads as private, permissive or another value.