Well, that sounds horrible.
Suppose, for instance, that the non-eligible officer committed massive crimes, and is arrested for those crimes. Of course, this whole situation is unprecedented, but how do you think that might affect the outcome?
And add to that the fact that many others conspired with him to commit criminal fraud so he could get elected?
That's a different matter. He can be prosecuted for any criminal acts.
But the fact that e.g. Bill Clinton was in fact a criminal did not render his official acts invalid. Same would be true if he was ineligible to hold office.
-- And add to that the fact that many others conspired with him to commit criminal fraud so he could get elected? --
Same as for the principal. If Pelosi falsified documents to secretaries of state, she's likely committed a crime, although each state has it's own criminal laws on points like that.
If a person is elected based on fraud to the public, the general rule is that the official acts stand even though the fraud is discovered and corrected. It's up to the public, not the courts, to run an honest system.