This TfL mayoral answer suggests there is no funding yet for any new stations on the Mildmay line, but coupled with the local plan below suggests the only thing likely in the next 10 years is Old Oak Common Lane, which would connect Richmond but not Clapham
Questions to the Mayor: Whilst we are waiting for further info about funding for West London Orbital, would the Mayor consider prioritising opening Old Oak Common Lane station, which would provide added integration with the future HS2 station and other local networks?
www.london.gov.uk
The adopted local plan is here:
I think the best you can hope for in the next decade is that Old Oak Lane gets built, which then provides a quick connection up to Willesden before doubling back South towards Clapham.
To make a station closer to OOC viable, it needs a lot of redevelopment to improve access, which probably includes moving Elizabeth Line depot elsewhere