Hello & Welcome to London . I'm from London but currently living in Sydney, so I know exactly how you feel.
Looking at places near Richmond, Kingston-upon-Thames is nice, very close to the Royal parks and close to town, with lots of shopping and water. Kew is lovely, as is Esher. There are loads really, it depends how close to the centre of London you want to be and if you need a tube line. The trains are good though, a lot of suburbs can get fast trains into the city.
Victoria is a good station, you can reach it from most places, so you can live pretty much anywhere.
I'm not sure how healthcare works for visitors, I expect you're the same as us in Australia - we get a medicare card and can use bulk billing doctors. I don't know if you'll need an NI number or if you can register for reciprocal healthcare directly with a doctor. I do know that you will get free healthcare though (I had an Aussie nanny when I was still in the UK and she had no trouble, I just can't remember the details).
With childcare I suppose it depends what you want. There are plenty of private nurseries around, no long waiting lists like in Sydney. There are also lots of Au Pairs and nannies around, much more so than Australia - with Au Pairs they'll often be girls from Europe who are looking to brush up on their English.
There are plenty of soft play centres around, I know of ones in Richmond and Chiswick, also a lovely soft play cafe in Chelsea (with video links from your table down to the ball pit). There's a kids cafe in Kingston too, where lots of mums meet up.
I hope you enjoy the UK . Take plenty of trips to Europe on the Eurostar while you're there and go to the West End too - those are the only things I miss. Aside from Marks and Spencer and the fanastic shopping [pines for Gap and Monsoon and supermarkets the size of warehouses]