Divorce is the only way for us after all he's done so I don't see the point in waiting- he said I'm trying to rush it through!
He doesn't want to rush it, because then he'll have to start making proper formal arrangements.
It's not okay for him to just see his kids when he wants, now that he's left. That has to be arranged formally, because he's unreliable and you can't hang around waiting for him. That's what he wants, for you to be stuck.
You need to put in an application for flexible working to enable you to do the drops and picks without him.
It could mean that you continue working from home after you pick the DC up depending on your job.
I work in HR and would be happy to help you with the wording of the application if necessary.
If he dropped dead tomorrow, you'd find a way to carry on with work and the kids.
Once he sees you can cope without his assistance, then he won't have anything to hold you ransom with.
When you mentioned something about his relationship with the OW being established... Who knows how long it's been going on. Maybe in his mind it is established. They could have been together for years, because nobody with half a brain would just leave after a couple of weeks with an OW.
I feel your pain, but your employers will be understanding.
Feel free to PM me if you need to regarding the flexible working request. There could be alternative solutions/working patterns which I might be able to help you with.