Bird nesting and tree trimming

Is now an okay time to trim a fir tree as far as bird nesting is concerned? I have a fir tree of sorts that has forked into two central limbs and I would like to trim off one of the the overhanging limbs but I've seen wrens flitting in and out so don't want to disturb any nests. I'm aware of the resulting brown interior that will be exposed.
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If you live in Derbyshire, as I do.
I did have a nesting box which attracted robins, but then my neighbour went and had some roofing work done in mid-nesting season, and they deserted the nest and have never returned.
I believe pigeons can lay eggs at any time of the year.
Hope this helps