I asked a similar question recently. Not about wood pigeons, though.

Certainly they would. It’s far less work to find or steal a nest than it is to build one from scratch. The bird could also be interested in the eggs in the basket, as some birds will eat undefended eggs; however from what I’ve gathered (in 5 minutes of Google searching) magpies are usually the aggressive egg eaters, not wood pigeons.

My final conclusion is there’s a 99% chance the wood pigeon is trying to steal the nest, and a 1% chance that it has been hunting down the magpie that ate his brother and is seeking revenge ;)