Apologies, I think I mistook what you were saying. I suspect it is more of differences in parish/county/town council signage.

I'm not 100% against tagging as a bridleway (the differences are pretty arbitrary in this case). But I'd lean more towards leaving it as a cycle path given that:
- It is part of the national cycle network
- Is referenced as a cycle path on the Town Council website
- The town side has questionable equestrian access