Sorry, no time to read the full thread but I’ve read your OP, really sorry you’re having to consider that.
We had our cat PTS last year. I also did lots of work experience at a vet’s so I know that if you don’t express an alternative preference they are normally put in a big freezer and then sent to the cat crematorium. In ours’ case we were given a towel to wrap the body in, I held him in my arms on the drive home and then stayed with him for a few hours. I believe this is because my mother opened for this choice. He was buried 24 hours later wrapped in a scarf of mine.
I haven’t ever been privy to an animal being PTS before and that gave me closure. I have seen (really sorry) cold and stiffened up bodies of family cats that were found dead, which were so much like a stuffed or taxidermist version of themselves it wasn’t too upsetting. Both have given me closure, but I did go through a bit of distress sitting with the one whose body I was in charge of while he still could just have been sleeping. I even, to my embarrassment, kept trying to wake him up.
If you can’t bury them pretty quickly, there are also some fairly unpleasant physical issues to deal with, as I said ours was wrapped in a scarf and before that a towel but when we transferred him to a box there had been some seepage.
I’d go with when it happens, know what you’re going to do with the body in advance and maybe sit vigil for a few hours and then put them to rest. It wasn’t exactly pleasant to cuddle the body of what had been my furry friend until fifteen minutes before all the way home, but I was glad I’d been there to and after the end, and even though I still occasionally look out of the window to see him doing his well-worn patrol it really did help with closure.
I don’t know much about children’s psychology either and my phone has a weird glitch which keeps erasing the text if I scroll up so I can’t check the OP, but it might even help them to be there to say goodbye and see the cat go to sleep. I disapprove of euphemisms like “pass away” for humans but being PTS really is very quick and painless.
Just please don’t keep any part of them. I came across the tail of one of my childhood ponies, who was shot without my knowledge or being able to say goodbye, in a drawer. Not fun.