I think you’re right that it’s been a problem with the French system. A couple of days after posting this, I got a notification from HMPO saying they had my passport, and it’s been plain sailing from there (so far!). My application was swiftly approved, then I got a notification saying it was on its way to print, and just yesterday found out it’s on its way back to me.
Let’s just hope my old passport gets back to my quickly too as that’s the one my French visa is in.