Hi
I live near Tring and know the trails pretty well. The Ridgeway is well signposted but once you get East of Princes Risborough a lot of the Bridleways change to footpaths. Personally I would ride them anyway, very rarely get any grief and the trails are excellent. If I were you I would do that but avoid doing it at weekends because some sections will be quite busy with walkers, especially around Coombe Hill and Wendover.
Some locals that walk on Coombe Hill get stroppy about bikes but go mid-week and you’ll see virtually nobody. I’ve never had any grief anywhere else.
In comparison to the footpaths the bridleways are a bit dull and will have been churned up by horses.