Good news! OpenReach turned up yesterday morning - no appointment and I was still asleep when they rang the doorbell so my conversation with them was a bit wonky to say the least. They asked permission to drill into my walls and prune my trees. I told them it was all fine by me but pointed out that I thought permission from the council was needed to prune the trees. They told me they'd go away and talk to their boss. I didn't expect to see them back anytime soon but they reappeared this morning to put the new line through the tree without cutting it and I now have a shiny new (ok, dull grey new) "OpenReach Customer Service Point" on my wall and I guess the contractor will be back to connect that up to the inside.
Seems like I was worrying for nothing, but also seems like they can put a fibre line through a tree or two. Yay!