Regarding Barnes (East Sheen and West side of Putney too). If you work at White City then the latest is that Hammersmith Bridge is open for pedestrians and cyclists. Barnes to White City is a nice flat cycle. If Hammersmith Bridge ever reopens for buses be warned that buses can take forever to get round Hammersmith bus station and to Shepherds bush, could be 20 minutes, could be an hour. If you find a house closer to Barnes Bridge station you can take your bike up and over the pedestrian bridge there if you can lift it up steps. North Barnes (Castlenau) would normally be pretty good for access to Hammersmith tube but it's very uncertain at the moment due to the bridge issues.
Wimbledon to White City isn't a great commute. The Overland from Clapham Junction is every 30 minutes and normally rammed at peak time.
You haven't mentioned budget, but you say state school. There is quite a split in the areas you mentioned of those who can afford private and those who cannot (or choose not) to.
I think Wimbledon is better for trains and tube generally, Putney may be an option too. Hammersmith bridge and flood risk are a bit of a negative for Barnes. East Sheen and Mortlake are in Richmond and between Barnes and Richmond town.
Traffic is bad (not helped by the bridge) around the south Circular because it isn't a dual carriageway like the North Circular and there will be a big development going in on the old brewery site in Mortlake, hopefully with a secondary school.
Have you experienced plane noise? That might be a factor for you.
State secondary schools get patchy in all these areas due to wealthy parents going private for secondary.
The London School Atlas is a great resource: apps.london.gov.uk/schools/
All the primaries in Barnes area are good or outstanding, some different approaches / sizes / mixes of pupils. Putney generally are good or outstanding too (you won't want Granard though)