It's a long time since any UK politician resigned on principle or on honor. I think the late Robin Cook was the last (over the Iraq invasion which he believed was illegal).
Private Eye has an endless stream of 'highly suspect' (since any suggestion of 'corrupt' may get HN into trouble, just to be clear, I'm not saying 'corrupt'.) activities of various government ministers, none of whom resigned or even came close - despite their misdeeds being exposed. Look at how many were caught in the expenses scandal - some UK politicians even tried to hold on while being convicted of fraud and other criminal convictions (with some initial success too, https://en.wikipedia.org/wiki/Nazir_Ahmed,_Baron_Ahmed). That's never going to fly in the US.
You really need to go back to Margaret Thatcher's cabinet before you can really find many examples of resignation on the back of government screwups.
I think a resignation is more likely in the US than the UK. And it's not at all likely in the US.