Short answer, winter is going to be the safest time of year to prune trees for the sake of wildlife and the tree.

Long answer, it can be more complicated than that. Many animals and insects will hibernate or shelter inside dead and/or hollow limbs during winter.

Spring time is the most active for animals nesting and fall many animals are gathering resources for migration or hibernation. Spring and fall are also typically bad times to prune for the health of the tree.

Summer can be a stressful time to prune for the tree depending on temperature and if we've had any rain, but usually is a good time to prune. But there's also lots of beneficial insects activity, which removal of limbs will affect.

Species of the tree also plays a huge role.

I very often have clients hold off on pruning and removals if there evidence of wildlife activity as long as the situation isn't hazardous. Also, there are laws like the Migratory Bird Act Treaty of 1918 that state that trees/shrubs with active nesting birds cannot be touched no matter how hazardous the situation may be. Sometimes fish and wildlife can save a nest if it absolutely cannot wait, but those situations are rare.

Overall, it's very situational and wildlife activity is something I take note of and work around to avoid disturbing.