To worry he won't do it

Insurance won’t pay out for something that’s your own stupid fault - like not applying for passports.

If he cocks it up it's not just him who suffers consequences, it's also the kids, who won't get to go on holiday (which presumably they're looking forward to), through no fault of their own.

And no travel insurance will pay out because passports weren't sorted on time.

Are they first passports or renewals?

However if he doesn't do it then there will be no holiday, no visit to my family

Do you think he would have done it by now remember if it wasn't your family you're visiting?

Of all the things you leave him to do why this one?!

Let him do something else

A grown man has said that he will apply for his own childrens' passports. He knows when the holiday is. OP has offered assistance, he has assured her it is in hand.

I don't think that dancing around creating false deadlines is necessary. Unless he has got some kind of extra special circumstances other than his male chromosomes, it's not your job to spend energy putting guard rails in for him.

Does he do that for you? Hover behind you making sure you don't need any help with the adulting? Thought not.

There are two outcomes here, one where he does it and all is well. The other where he doesn't, and he will feel the natural consequences when he is at home staring into the freezer and wondering what to make his DC for tea and lying in a dirty bath that he forgot to clean, while you're on a sun lounger wondering why you married him in the first place.

This.

Stop micromanaging him! If you want to do it then do it, if you want him to do it, let him. No reminders, no gentle nudges, no fake deadlines or other tricks.
I say this as a control freak myself and totally understand the urge to remind and worry, but honestly it’s just going to cause you both resentment. There is no point sharing out tasks with him if you’re still going to hold onto that task yourself.
He is an adult man. I expect he meets all his deadlines at work without your help.
If he fucks it up he will have to deal with the consequences.

I mean, if he doesn't do it, you could still go and leave him with the kids but I understand that doesn't fill you with joy...
I'm a big fan of people getting natural consequences to things but when it's a trip like this I can see why you are anxious.
But then on the other hand - if he doesn't learn now, when he's a grown adult with kids, then you and the kids will be putting up with this his whole life. As you say, he can sort car insurance so he's perfectly capable of doing this.
Really tricky one. Could you say to him that you want the passports back by September?

It's called "weaponised incompetence" and is a deliberate tactic which absolves them of being asked to do anything.

Introduce an artificial deadline - holiday company need details be end of August.

Honestly...just leave him to it. If he doesn't do it then he is responsible for the consequences. Sometimes people need a wake up call and if he forgets to do it then it's all on him to fix it.
If you're flying abroad then passport details will need to be entered for all travellers so do it the month before and ask him for the passports then. If he's forgotten then HE has time to sort it.

I'd get some good insurance and leave him to it.

PS, he doesn't forget to clean the bathroom....

I wouldn’t, I’d make sure he’d done it. Sometimes there’s a hold up and with a holiday booked there’s no point in waiting and hoping he’ll get it sorted. My husband is a terrible procrastinator and leaves everything to the last minute and I wouldn’t want to leave something so important for him to do.