If a mother doesn't want her ex to be part of their children's lives, it's pretty easy for her to make that happen. And it's very, very difficult for fathers to get meaningful contact back.
It really isn't as simple as just filling in a court application. First of all you need to locate them (as in the OP's case the ex moved away with the kids leaving no contact address). Court process for that, and if you're lucky they'll be located. Assuming the mother doesn't then claim domestic abuse to try and prevent the contact details being shared.
Then you can apply for a CAO. But by then the kids will have lost regular contact with their dad (and even if the mother hasn't claimed abuse), the court is unlikely to immediately award unsupervised contact to start with. It'll be a gentle reintroduction - occasional supervised contact, probably at a contact centre.
Then the mother has to actually comply with the contact order - how unfortunate DD or DS happens to be ill on contact day. If you're lucky you'll find that out in advance, otherwise you'll be waiting in a contact centre (that you've paid to use) to find out you've been stood up. That'll probably happen repeatedly, and you'll never build up the supervised contact to a point where unsupervised contact will be granted. Sure, fathers can go back to court to try and get the order enforced, but that's another battle in itself.
My guess is that in this case the OP's partner's ex has come out of the woodwork either because she's decided she wants some child free time, or because the kids have nagged her into submission about seeing their dad. Seriously doubt it's her conscience alone that's caught up with her.
Ex is still being manipulative and controlling here though. After 6 years though, the kids must both be old enough to use skype/facetime/zoom by themselves. So there's no reason for the parents to have direct contact with each other - just video call the kids.