Tree cutting found nests

I feel awful!!! I moved into a cottage with a wildly overgrown garden . I've had issues with rats and have had to remove a huge pampus grass and butterfly Bush from the front. This week I tackled the bad garden...a massively overgrown cherry tree and elderberry tree. These trees though are home to some lovely birds,  and iv ruined their home. I found multiple nests and they even landed and sat on the nests when I stopped for lunch. What can I do ? Will the pigeons settle for a bird box? According to neighbours the pigeons have lived in the trees for a long time, it's actually keeping me up at the thought I've destroyed their home....its wood pigeons I believe and I see blue bits coming and going but the pigeons are residents, we'll were.

Any advice on how to help them adjust to being homeless,  I've kept two nests but not sure what to do with them.