If a politician refuses to resign after being voted out of office there might be a Constitutional matter to settle. What the British Constitution says about the case will have to be taken into account. The decision about what to do will most likely depend on how they interpret the Constitution.
This does not create a Constitutional problem in the U.K. If Corbyn uses this time to change his ways and policies in such a way that he instills more confidence in the people. He is not required to resign after the vote and if he chooses not to that is his choice.
British Labor Party leader Jeremy Corbyn refusing to resign after losing a no-confidence vote does not create a constitutional problem in the U.K. Corbyn will no longer be an effective leader, nor will he remain popular with many voters. However, Great Britain will not have a constitutional crisis over this issue. Eventually, Corbyn will be voted out of office if he becomes that unpopular.
Supporters of Mr. Corbyn have called on his critics to either shut up or challenge him in a formal leadership contest. Under the Labour Party system, the final choice lies with party members and supporters, and Mr. Corbyn’s allies, who include influential labor-union activists, are confident that he would win re-election, despite some indications that support for him is weakening.