What would you do?

The problem is that he is free now and not likely to be in autumn.

I have the same type of hedge. And similar problems.
I'd leave it until Autumn
Why?
1. It's going to grow more energetically after the cut 'cos it's growing season
2. Wildlife

Ring around and get yourself booked in for Autumn.

whywhywhy

Leave it until autumn.

This.
At least you’ll avoid a potential onslaught by ornithologists

Our hedge is a mess but not getting it cut until September due to bird nesting.

Would it be that bad to wait a month or so?

I’d probably go ahead with the trimming since you haven’t seen any birds nesting yet. But maybe just check the hedge carefully before they start, to be sure. If you spot any nests, you could ask the trimmer to avoid that part for now.

We have the same wild hedge on one boundary. Allowed to do it's thing until late October when severely cut back and we have harvested all of the blackberries of course .
I would keep your slot but as suggested maybe not too harsh until later in the year.

Leave it until autumn.

Can he take a look at each section as he cuts to check for nests?

The ‘someone’ if suitably experienced, should be proficient in checking for nesting birds, my landscape gardener friend does in very thick hedging.
So, I’d make sure they are capable, if so trust them to do the job.

Looked it up - nesting season is from February - August.

Perhaps a light trim? If necessary and book for beginning of September, then it should look tidy all winter.