The power of the Privy Council lies in it's executive committee, known as the "The Cabinet" that thing chaired by the Prime Minister we call the democratic government. The rest of the privy council membership is mostly a bauble for past cabinet ministers with some royal flunkies and bishops and the like. It's mostly vestigial, like knightly orders, but with weird exceptions like it includes the supreme court for overseas territories.

This isn't to say such things can't happen but it would not be through a recognised legitimate procedure "with teeth" but as a constitutional crisis where precedence, tradition and law has gone out of the window and whatever side wins is through primitive power/confidence dynamics. There might be rulings of lawfulness in one direction or another but as a postfacto figleaf downstream of victory rather than as a real judgement.