Birds will start nesting and rearing chicks from about March until August time. If you are going to remove the hedge then you should do it immediately or leave it until early autumn.
From personal experience, the birds will be using your hedge for shelter and for roosting all the year round. Immediately after it is removed the birds will be making extremely loud and rude noises at you! But they will soon get over it and find new haunts within days.

I sympathise. We had someone in to cut ours last year as we have three, privet, beech and unfortunately leylandii. It really was money well spent, and of course he took all the clippings away with him.

Hi, I agree, do it now before it starts into growth, or you will end up having to cut it again, Derek.

Afterthought - fences can and do blow down in gales - hedges don't!

Yes, the saying swings and roundabouts, springs to mind, lol, Derek.