No Neil, it is not.
Since my last posting I have checked my sources for this area, and can confirm that it is a "Permissive Path".
It is an "Other Route with Public Access". Note the term "access".
It is not shown on the Definitive Map as a legal Right of Way.
Allow me to put my former local authority planning policy hat on.
The local authority map is not as definitive as the name suggests. Local authorities have been legally obliged to record rights of way on the definitive map since 1949. This process is not complete, however, and many rights of way are still not formally recorded. I would estimate 10% of rights of way are still not listed on the definitive map. Under the 2000 Countryside and Rights of Way Act, there is a cut off point of 2026 to get paths which have been in use before 1949 listed on the definitive map.
The 1980 Highways Act allows for the presumed dedication of a right of way provided it has had uninterrupted use over a period of 20 years (presumed dedication can also happen under common law... but let's not complicate things too much). However, this presumption can be overridden if the landowner never intended for it to be right of way. This might be evidenced by the land owner physically interrupting the right of way (installing fences, walls, etc) or with signs at the location explicitly stating the path was private property and not for public use. A statutory declaration stating the path is not intended to be a right of way can also be made with the local authority concerned
So regarding the status of the bridge as a right of way two questions need to be asked:
1. Has the public been able to access the path across the bridge for an uninterrupted period of 20 years? Given the bridge has been in situ since the 19th century I'm guessing the answer to this question is probably yes.
2. Has the landowner (i.e. Network Rail) made it clear they do not intend it to be a right of way? Here things get a bit murky. The bridge is obviously private property, but this alone does not prevent right of way access. There may well be signs at the location stating it is private property and not a right of way. Network Rail many well have made a statutory declaration to the local council. This might go someway to make the case it is not a right of way. However it could be argued this is contradicted by the active encouragement of pedestrians to use the bridge (albeit upon payment of a small toll) and its more recent explicit inclusion on the Welsh Coastal Path.
It is up to the local authority to decide this, although the landowner would have right of appeal to the courts if it were unhappy with the decision.
Regarding the costs to the local council: It is landowners who are responsible for the maintenance of rights of way to ensure they are visible, safe and accessible. However, landowners are able to claim 25% of the costs of structural features to maintain access (although many councils until recently funded 100% of the cost). This usually applies to stiles and gates, but I guess it might also apply to a foot bridge (albeit a rather large one).
Given this, and the fact the bridge exists and Network Rail have to maintain it regardless, it might actually be in their interest to make sure it is listed on the definitive map as a right of way. That way they would legitimately be able to claim some of the costs towards it's maintenance and the council would have to pay (although if they want £30,000 per annum towards the cost of the bridge they would need to demonstrate that the footpath part of it costs £120,000 a year to maintain). I'm guessing the current arrangement is voluntary on the part of the council?
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I'm aware of that, but they certainly give an indication. However, another poster seems certain it is not a public right of way.
As I've just described, there is no such thing as certainty when it comes to public rights of way until either a court has made a judgement to that effect, or until the 2026 cut off is reached for additions to the definitive map of paths with access prior to 1949.