Hi OP,
I moved to Europe so not as far as Australia, but I do recognise some of the struggles of settling somehwere new. You have said that you are a SAHM and I can see how that might be isolating in a new environment - if you're willing to stick it out for a bit then perhaps you really need to work on your social circle. I moved recently and making new friends in your thirties is daunting, but you may have the "ready made" option of mum/toddler groups that could help you.
If you were willing to stick it out, are you willing/able to work/study there? This could integrate you further and make you feel far more settled.
However, the bit that made me double take was your partner saying "take the kids, go back to the UK". I cannot imagine a partner, let alone a co-parent saying that. I live/work/socisalise with expats and, rightly or wrongly, I am immediately suspicious of men who live in different countries from their kids - unless there is a clear, obvious and burning financial reason when he is providing for them/his family.
Other PP have said that he is effectively giving you permission, which he may withdraw later, so this may be your chance to go. You also call him DP and not DH, so honestly, this could be your best chance to leave with your kids. Should you marry, Australia is famously difficult to leave with your children if the other parent does not consent - there have been a number of threads about it on here.
Can you access therapy? I think you need to work through the emotional implications of him being apparently willing to let you all go, then take him at his word before he changes his mind.