Neighbour felling tree with nests directly under it.

My neighbour is felling a huge conifer next week. It's more than twice the height of our houses and despite loving trees, I can see it needs to go at some point. I can't see any nests in it this year, but only 30cm underneath the lowest branch, and only a metre or so from the trunk, on top of my shed, is a wood pigeon nest. The only thing protecting it from the elements is the conifer. There's also a collared dove nest well underneath the conifer canopy in another small tree in my garden. Both have birds on and have had for at least two weeks. I've told the neighbour about the nests and they said the people booked 'will still do it.'  I'm pretty sure it's illegal as there's no way to do the work without disturbing the birds or hurting them, and have nicely told the neighbour this. Luckily, I'm off work that day so am in a position to do something and they need access to my garden to remove the bigger branches. 

Are these nests protected by the Wildlife and Countryside Act?   Thanks.