You've posted in 'Music', Dmbhunt!
I carry out surveys on parts of the National Cycle Network, so I spend a lot of time on cycle paths. The vast majority of them (including 'dual use' footpaths) have 'Give Way' markings on them where they cross side roads. In such cases turning traffic clearly has priority.
However if there are no 'Give Way' markings then (as footpaths and cycle paths alongside roads are still part of the public highway) 'straight on' traffic [i.e. cyclists] will have priority over turning traffic.