It seems that it is up to the local authorities whether we can ride on bridleways or not, looks like we are slowly being muscled out of places where we can ride.
See extract below. Not sure if it has been updated recently.
http://www.legislation.gov.uk/ukpga/1968/41
Riding of pedal bicycles on bridleways.
(1)Any member of the public shall have, as a right of way, the right to ride a bicycle, [F87not being a mechanically propelled vehicle], on any bridleway, but in exercising that right cyclists shall give way to pedestrians and persons on horseback.
(2)Subsection (1) above has effect subject to any orders made by a local authority, and to any byelaws.
(3)The rights conferred by this section shall not affect the obligations of the highway authority, or of any other person, as respects the maintenance of the bridleway, and this section shall not create any obligation to do anything to facilitate the use of the bridleway by cyclists.
(4)Subsection (1) above shall not affect any definition of “bridleway” in this or any other Act.
[F88(5)In this section “mechanically propelled vehicle” does not include a vehicle falling within paragraph (c) of section 189(1) of the Road Traffic Act 1988.]