Other posters have pointed out that even if you ignore environmental considerations, and somehow get round the land ownership problems (a lot of problem trees are on private land - imagine the expense of having to negotiate with the owners of hundreds of properties, and the difficulties of dealing with those who won't cooperate), you still have the issue that some of those "damn trees" may be holding embankments and cutting slopes together with their roots. Just chopping them down could result in some pretty disastrous consequences. Given the uncertain internal construction and condition of some of these structures and the variability of tree roots it may not even be possible to tell if removing the trees would destabilise a given structure or not until you do it. Very extensive, expensive and disruptive stabilisation work might be needed if you removed all the trees and even trying to put a figure on such works on a country wide basis would probably be pure speculation.
Of course some flat areas where the trees are on railway land may be good candidates for removal, but as a general solution I'd favour things like fitting decent sanders as likely to be more cost effective and quicker than any total, wholesale network-wide tree removal (which might not be either legally possible or permitted by the government anyhow).