Any C, D or U roads listed in any part of the UK should really be tagged using the highway_authority_ref tag, as these are only used by councils to classify their road network (and as such many of these numbers may be duplicated across the country). Motorways, A-roads and B-roads are unique numbers that are widely signed. Of course there are always exceptions where a road may not be signed to discourage traffic or signed in a different way to provide clear information to drivers (such as here) but that doesn’t mean the number is less important to tag using the ref tag.
As an example, the A2690 in East Sussex only has its number mentioned on a couple of signs despite connecting two large towns, because the end destinations of the A21 and A259 are signed as more convenient destinations. This doesn’t mean the road should have its number moved to the highway_authority_ref tag on this basis.