I am pretty sure that if the locals really wanted they could probably get TPOs put on some of these trees and make lots of trouble for NR.
If you mean trees on railway land, then not really, as Network Rail is a statutory undertaker (i.e., it has responsibilities under statute law, not that it has to arrange everyone's funerals), like for example the electricity distribution network operators. There are exemptions in the TPO legislation in that the LPA's consent is not required for works to trees by statutory undertakers on operational land in order to comply with their statutory obligations, even if there is a TPO in force (which is unlikely - LPAs are unlikely to bother making a TPO that cannot be enforced).
There are other legal considerations relating to vegetation on railway land - I believe Tring cutting is a SSSI (chalk grassland) so anything there would need to be agreed with Natural England - and the presence of bats or birds nests which have legal protection may need to be taken into account, but the TPO legislation is largely irrelevant.
Even with TPO'd trees not on operational land where NR couldn't claim the statutory undertaker exemption, they could still apply for consent to pruning/felling and if they could demonstrate why the work is required, and agree to replanting if the LPA requires it, then the LPA could allow them to go ahead.
None of which is to say that NR shouldn't consult with the LPA & local residents about works to trees, nor ignore good practice in environmental stewardship.