It is perfectly legal to shoot Wood Pigeons ( as opposed to Wood Pigeon's, whatever they may be), although you would have to be mindful of the distance from a main road, if using a shotgun ; a "Pedretti" fully -moderated (silenced) .410, 3 inch Magnum is MOST effective,in my personal experience.Or you could use an air rifle, of course.Jays too,are legitimate quarry, and as for Grey Squirrels, well .....!!!
          Want me to do it for you ??