^This is from a Ramblers Assoc. website. It's important because many LDWA members in my experience are just not aware of the legal basis of PROW's.

Who owns the paths? The surface of the path is for most purposes considered to belong to the highway authority. What this means is that the authority owns the surface of the way and so much of the soil below and the air above as is necessary for the control, protection and maintenance of the highway. The rest normally belongs to the owner of the surrounding land.

This supports my view that the golf course does not exist on the PROW that crosses it.

A footpath which is a Right of Way is the Queen's Highway and as such the surface at least is not part of the golf course. I don't believe that there is a generic law against playing golf on the Queen's Highway ( although Motorways are a a clear exception) but clearly the onus is on the golfer not to obstruct the walkers/riders right of passage and smacking him/her one with a 100mph golf ball would almost certainly be classed as an obstruction, if nothing more. The same (but with more law attached) applies to the loonies who shoot pheasants next to PROW's.

Google "who has priority when a footpath crosses a golf course?" and you'll read several examples of rules and notices that indicate that walkers are given priority. For example, "Paths run alongside and cross several holes on the course. Priority must be given to members of the public. Players must not play shots until footpaths in range are clear. Tee signs have been erected at effected holes." Perhaps not every golf course with a footpath across it will state or enforce this, but it's a reasonable expectation.

It's an offence to drive a motor vehicle on a road recklessly, carelessly or inconsiderately and the definition of road includes footpaths and bridleways. I accept that doesn't spell out who has priority and it doesn't mean that vehicles and walkers can't ever share a road, but it does mean you can go to the police if you believe an offence has been committed.

Just out of interest anybody know who has priority when a private track, golf fairway, runway crosses a public footpath / right of way?

For golf fairway, I would expect that it is a golfers responsibility to make sure that they don't hit a golf ball across a public right of way when somebody is crossing.
I would also expect that traffic crossing to / from private land on either side of a public right of way should do so in a safe manner, giving way to traffic (foot or if legal horse/bike)

But I'm not sure?