Try Aviva and Quinn anyway (if you haven't done so already). You never know, they may say yes. If they do say yes, you should also clarify what cover you'll have for conditions that develop while you're on Plan P (or equivalent) after you upgrade back up to a private cover plan.
One other thing to bear in mind is that even if Aviva or Quinn say yes today, they may not necessarily honour that if you're joining, say, next year. So it might be something you'll need to check every year whether they say yes or no.