Unfortunately, removing the highway=unclassified tag from part of Hedley Avenue effectively removed it from the map. Private roads are still roads, so this tag should not be removed. I am not sure what "no highway status" means, but Hedley Avenue is in OS Open Names and has a Designated Street Name USRN, so it will also be on OS maps and derived products.

Also, I can see from Bing's street side imagery that the entrance to Hedley Avenue is signed "private road". This means that it is privately owned, not that access is private. In this case, ownership=private + access=destination is probably the best tagging.

OSM data is used by navigation software, users of which would reasonably expect to get them to the gates of the premises to which they are travelling.

You may find the documentation for these tags on the wiki helpful:
https://wiki.openstreetmap.org/wiki/Key:highway
https://wiki.openstreetmap.org/wiki/Key:access
https://wiki.openstreetmap.org/wiki/Key:ownership

I have restored Hedley Avenue and added a little more information in https://www.openstreetmap.org/changeset/150765919