Tbh, windmill our suburb isn’t so different from Wandsworth (I’ve lived there too!) in that it’s an older, established suburb, we have good public transport, a village feel with cafes, wine bars, 20 -25 mins to the cbd..
We looked at 150 houses (!) when we were looking to buy - we didn’t know which suburb we wanted to be in, what we could realistically afford and what kind of house we wanted. We still ended up with a head rather than heart buy, but it’s fine and housed four, sometimes five, of us quite amicably through the endless lockdowns! It’s close to where you were looking in Cheltenham.
Ive just seen that you’re looking at Moonee Ponds and South Yarra. South Yarra is a fab choice if you like a buzzy life and being close to the city. Good train links to everywhere too. We were very close to there when we first moved here - our adult kids loved it! I don’t know Moonee Ponds so well.
Another place I have friends is Port Melbourne. Lots of apartment living if that’s what you’re after and close to the water and trams (and lovely Albert Park). Might be worth looking at if you haven’t already, although I don’t know how expensive it is…
Good luck with your move - I hope you find exactly what you’re looking for and are very happy here.