Despite those maps being "© Crown Copyright and database right 2024" I believe that I've seen similar data released under OGL (and hence licence-compatible).
I'm a bit confused by what I see at both "Natural England - Open Access maps" and "CROW Section 4 Conclusive Maps" linked from there - locally to me only some of what I thought was "common land" is listed as such, but there's extra sections of "access land" that I wasn't expecting!
Nothing listed in the Woolton area though.