My heart goes out to you - however, you are probably not going to have an option here and your best bet might be to work out how to get on with life. I am also in a mixed marriage, had kids, had a DH who refused to move (changed his mind thank heavens!) but my kids were grown and gone before all came to a head. My husband also used the "but then I would be depressed" argument in one of our discussions - he has since apologised and now living in UK seems to be like a dog with two tails!
First, take yourselves off to marriage counselling - you can talk around a subject (and I do mean around) without coming to a decision when you do it on your own but upping the ante a tad and getting third party intervention to help you both focus on what is a liveable compromise situation is going to be a good starting point.
Second, your mental wellbeing is at risk here and your GP can help you - situational (exogenous) depression is a very real beast and if it impacts every day of your life, as it seems to be doing with you, it can be "treated". A GP can arrange for a mental health plan for you and some time with a good CBT or ACT therapist might help you with skills which can help you get through every day without wanting to slit your wrists. Please seek help. You've done all the usual things that get trotted out to help you "belong" but it is OK to admit that you don't belong despite having tried (I tried hard for 32 years, nothing happened!)
The family court in Aus is not helpful in this regard - if you decide to split from your DH and return to Wales there is no chance that the court would let your kids leave if he said they could not - none at all, zip, zero, nada! He holds all the cards here in this regard so you are probably stuck regardless.
As I said, I have been where you are - the question I always came back to is "is this the man I want to grow old beside?" the answer, of course, was yes (Ive had him for 42 years, not going to train up a new one) so my rationale always was that life in Australia with him was less worse than life in UK without him so we compromised as best we could and that meant he ponied up for trips home when and as I needed them. It was not ideal and my mental health really did suffer although I was well equipped with strategies to cope with the depression that overwhelmed me but the thought of being without him kept me going. So I guess that is the question you need to be asking yourself and then working out how you can make all this work for you.
Just as a final nail in some coffins - even if he did say yes he would move, could he???? The visa situation for non European spouses is somewhat draconian at the moment. If he has some UK ancestry (grandparents or parents) then he is good to go with an Ancestry visa otherwise a spouse visa is going to be very difficult for you, as a young mum, to contend with (you will need an income of £18.5k pa to sponsor him)
So you have my sympathies, I hope you can get some support to help you through all this - it is possible to live in a place you don't want to be but it takes a lot of work. (((hugs)))