It's the same answer as for Footpaths. It depends on whether the Cycle path is located in a Designated area such as a SSSI/Nature Reserve/Area covered by a Landowner Agreement. If the answer is yes, then you are required to provide Proof of Permission. If No, then Permission is Presumed.
Have a look at http://www.follow-the-arrow.co.uk/resources/ukmap.html for links and a explanation of MAGIC. Which allows you to filter Designated areas.
And https://www.gagb.org.uk/land-agreements.php for a list of Landowner Agreements and those who have Banned Geocaching on their land.
Deci
My post is my personal opinion and as such you do not have my permission to quote me outside of these forums!
Dave
Brenin Tegeingl
Formerly known as Mancunian Pyrocacher on GC