Thank you Jimmy for responding to my note. 

I did not make it obvious that there is a routing problem for both foot and car access that needs resolving.

Broomfield Close is not tagged as a private road despite cars only being able to access it via Broomfield Hill, so presumably it's also private for cars?

Buckinghamshire Council, as far as I can tell, have not designated this footpath, Broomfield Close or Broomfield Hill as public rights of way, although there is a designated bridleway that joins Broomfield Hill just to the north:

https://www.openstreetmap.org/way/89886846

Are you aware of any signs or gates that may act as a barrier to pedestrians, cyclists or horses attempting to route through this road?

In theory, the owners of the private road should be able to revoke public access, if it's permissive. Otherwise this may be a problem on Buckinghamshire Council's part where they haven't secured public right of way through a private road.