I fear the action they'll take will be to close the path on 'safety grounds'
you could be right.
"
Traffic Regulation Orders
A local highway authority (also known as a traffic authority) can make orders under the Road Traffic Regulation Act 1984 to permanently or temporarily prohibit the use of any public path which falls under its authority.
Anyone who uses a path contrary to a Traffic Regulation Order (TRO) is committing an offence and could be liable to a significant fine.
The rights which are shown on the Definitive Map and Statement do not change because of the TRO but they are subject to the restrictions stated in the TRO.
Temporary Traffic Regulation Orders
Under the same legislation a local highway authority also has the power to temporarily close a public path for reasons of public safety. This can be because works are taking place on or near the path, or for other reasons such as an unsafe bridge.
Temporary Traffic Regulation Orders (TTRO) can be made for footpaths, bridleways and byways for up to 6 months initially (18 months for all other highways). If more time is needed, the local authority can apply to the Secretary of State to extend a TTRO. It is possible for footpaths, bridleways and byways to remain closed for years whilst development or repairs are ongoing.
Where possible, the notice advising of the temporary closure should include details of an alternative route. But this is not a legal requirement so sometimes you may have to find your own way of reaching your destination.
Once the TTRO expires, the path should be reopened along the original line. The surface should be repaired to the same standard as it was before the closure (or better). "