You’re already skating on very thin ice.
Under the EU birds directive ( of which you are still signatories ) you are obliged to provide protection for all birds during the breeding season, for woodies that’s practically all year round.
As a pest species they can be controlled, but it’s questionable if the UK’s rather liberal interpretation of what’s permissible, effectively any pigeon anywhere anytime whether or not damage is actually being done, would withstand a challenge.
If they are a designated quarry species, they would have the protection of a closed season.
Shooting during the closed season would require a special license and could be limited to particular areas and methods.
If you want to see how messy dealing with pest bird species can become, have a look at the Irish NPWS regulations.