If wood pigeons came from some tropical region of the world, we would marvel at their beauty. I still do, fabulous creatures.
That aside, as I always say - very nice with thick gravy, although we had woodies baked in a bed of cabbage a few weeks back - VERY nice despite what you (and me) might think.
Squirrels are introduced and officially vermin, as are muntjac. Also nice with thick gravy. (Squirrels more so fricasseed as per various US recipes, where eating squirrel is common. Sort of half way between chicken and pork for me).
If deer really are a pest, invest in a cheap electric fence.
I used to work with a guy who ate (at least once) anything that was edible, particularly roadkill or what had come to grief on power-lines (loads of those where we worked). He had eaten crow - and didn't recommend it; I suspect that maggies would be similar.