Depends whether the nest is currently occupied, you need to check. Blackbirds often have 3 broods a year, may even attempt 4. A clutch of eggs takes about 13 days to hatch, and the young will fledge after another 13-14 days, so its a case of catching the in between time when one lot's gone and another lot of eggs haven't yet been laid. Try not to disturb the nest too much when checking it or the adults may desert any eggs or young within it.

id cut up to about 10 feet of it then do as bamboo sais .

Not before end of august as far as farmers are concerned with hedge cutting
but with careful observation you may see when the birds leave the nest and not return

I'd try cutting the bottom part,so it clears the lawn but avoids disturbing the nest too much. But even then there's a danger the blackbird might desert the nest - possible worth a try though? Could you use hand shears rather than risk the noise of powered trimmers?

Leave alone 'til they've fledged, simple as that

Beware, It's actually illegal to interfere with a birds nest when they have eggs or young.

Thanks for all your comments. I am fully aware that I cannot interfere with the nest which is why the hedge hasn't been cut for years. It's not a hedge as such just a huge bush 7ft tall and 10ft long.
The nest seems to permanently occupied but I will wait until the end of August as Pamg suggests.

Thanks once again.

your welcome .

Holly, can I ask what the hedge/bush actually is?

Well Bamboo, at the moment according to the law it's a blackbirds hedge. LOL.

Bamboo, I don't know what it's called, it's not privet but it is as common as. Very tiny dark green leaves.

Sounds a bit like box?

Need a photo really...

Could be. Don't worry I am leaving it for now. Thanks.