I believe that for all accommodation bridges, underpasses, level crossings, it is the railway's responsibility to maintain the infrastructure. The duty will usually extend back to the original Act of Parliament under which the line was built.
It would be a good idea for NR» to buy-out the rights of landowners who no longer use bridges such as this, and demolish the structures, so that they no longer need maintenance.
I think this concept has not had an entirely positive reception in relation to historic railways assets. Demolition (or filling with concrete on unused routes) hasn't always found favour in the courts.