If a bank in these territories fail, the responsibility for handling the situation would primarily fall on the local authorities and regulators. They would be tasked with finding a solution, whether through restructuring, liquidation, or other means.
Unlike larger countries where the government may intervene to bail out failing banks, the UK government would not be obligated to step in and provide financial assistance to banks in Jersey or IoM. These Crown Dependencies are responsible for their own financial stability and crisis management.
how likely do you think it is that a bank in those countries will go bankrupt? I doubt they will, these are well managed jurisdictions for wealthy people.
Banks like that are often very solvent with healthy balance sheets, but the smaller ones are often at the mercy of their correspondents to actually operate. It's more likely that AML/KYC failures (leading to loss of correspondent relations, leading to inability to transfer money) cause them to shut down than being illiquid. Banks in places like Jersey, Guernsey, and Isle of Man have generally speaking very stable correspondent relationships thanks to having rigorous AML/KYC frameworks in place.