It is not a driveway because (as you observed) it serves more than one property, whereas in OSM a driveway is defined as minor service road leading to a specific property. For the same reason it access cannot be private – since access is needed to reach destinations further down the road.

It is common for early 21st century culs-de-sacs to be designed in this way, with the last few houses on each twig of the tree being constructed to a lower standard, with shared surfaces and no turning circle at the end. These public highways on which the normal rules of the road (including speed limits) apply.

These stretches tend not to be adopted by the local highway authority, but that is just a matter of who is responsible for the maintenance – not an access limitation. Hence, you cannot deduce access restrictions based on land registry plots. Defining access as private would mean that the first plot of land that any unadopted cul-de-sac crossed would be a ransom strip. 