A little late to this, but I've always found the minor league structure interesting, so I'm happy to stumble across this now.
I think it will happen eventually, it makes too much sense. I'd say you can circle MiLB's 2031 AAA Opening Day as your first opportunity to see the Richmond Flying Squirrels play as a Nationals affiliate. There are conceivable (convoluted?) ways to map it out before then, but as others have pointed out, it takes a lot of moving parts, so it just isn't very likely anytime soon. The current license agreements between MLB teams and affiliates expire after the 2030 season, and expansion should be ready to go by then. That seems like the perfect time for this to happen.
When they came close to moving to Richmond during the MiLB reconfiguration in late 2020, Rochester was rumored to get the demotion, and would've been the Blue Jay's AA affiliate. But I don't think it would be easy to force a demotion on Rochester while they're in the middle of those contracts. And I don't think a demotion would be necessary with expansion pending, so there's no reason for Rochester to agree to that right now.