It is the tree owners responsibility - but BT will be proactive and deal with it, if they can, as it will cost less than a line repair.
From experience though, the council will just say "so what, not a problem ..." and ignore it - or they would around here. Recently we told them every month for 6-9 months that a tree (on land they are responsible for) was gradually toppling over - they even wrote to me saying it was safe. Then one day, as forecast it came down - narrowly missing several vehicles and blocking a main road! Councils in general tend not to tackle small issues until they become major problems with a massive increase in cost.
M H C
taurus excreta cerebrum vincit