Thanks for your detailed response.

From what you have said, I think the correct tagging on The Drive would be ownership=private + access=yes

Pine Trees Drive should be OK as highway=residential without any ownership or access tags.

Dukes Ride and Georgian Close, if owned privately, would probably be ownership=private + access=destination

For personal use only, you can find out whether a road is publicly or private maintained from findmystreet.co.uk, but the licence for this is incompatible with OSM.

Updates to OSM can take a couple of weeks to propagate to Komoot, but I've noticed some quicker updates recently (it's my preferred site for planning running routes).

It might be worth opening a discussion on this in the OSM community forums, as other mappers more experienced than me may have other suggestions.

In the event of a dispute,the Data Working Group (DWG) can arbitrate as a last resort.
https://wiki.openstreetmap.org/wiki/Disputes

I hope that's helpful to you. Good luck and happy cycling.