If you know which area you want to rent in, find the revelation fb mums page. Meanwhile, you can ask them to get your a relocation manager/ company, they can do all the hard work on finding the flat.
The fact that they'll pay school fees is excellent, although not necessity. If they'd do that, and you don't need it, or end up paying the public school fee for a visa family, maybe they'd give you money towards more flights? Given private schools can be £15-20k a year, that's a lot to play with.
Given that you've both had a rough time, this could be exactly the bonding experience you need.
Be warned though. The kinds of area you'll be looking at, the unexpected costs really pile up.
There are hardly any "free" GPs left, which means you'll be paying £15-20 a visit to cover the gap, every time you need to see any kind of consultant it will be £150 after rebates, you need to make sure they include health insurance as without it, as a temporary visitor you'll be stuffed.
I understand that not everyone needs health services all the time, but dental, optician, all predicated on having health insurance. Most has 3 months - 12 months waiting before they'll cover half the things.
Meanwhile, whilst being incredibly general, it's not uncommon to spend £30-£50 a week for petrol, and you won't get a coffee, smoothie, and a couple of snacks for less than £15-20 in many parts of Sydney.
Unless you're planning to Aldi/Lidless your groceries, they are extremely expensive - depending on the food you like.. I'd say a "normal" shop for 1 adult and a hungry child might be - £80-100 a week. I'm assuming you'll be working a lot, and not have time for picking reduced stuff at all times etc.
Anyway, having said all that, and despite the fact that I'd rather not be here, I'd say go for it.
Good luck!