How do you cope with a large hedge?

That act makes it "an offence to intentionally damage or destroy the active nest of any wild bird", and hedgerow management rules apply to agricultural land, not domestic hedges.
Obviously, if you know there are birds actually nesting, you may not want to disturb them.

You’re not supposed to cut hedges between March 1st and August 31st as birds nest during that time and it is an offence to disturb them under The Wildlife and Countryside Act 1981.

I haave a neighbour who kindly attends to my hedges, and the one I have that sounds much like yours OP is severely cut back in the early autumn.

We too have a long hedge that separates us from our neighbours. Their garden is a jungle. We have the hedge cut twice a year. February and October / November. Never during nesting time. As this is the time when the hedge grows most, it looks untidy. We just have to live with it until the autumn.