An update: The dishwasher continues to work brilliantly, even though Mr4 is constantly pressing all its buttons and opening the door.
Sadly, our oven stopped working right after the dishwasher went in. Not to suggest one had anything to do with the other, just to say the timing could have been better.
So we've been oven free for a fortnight or so. When your kids love lasagna and oven fish this is an issue.
Seems to be the element most likely. But I'm going to pull it out (it's a wall one) and just get a new cheapo replacement rather than repair. All the markings for the controls had been rubbed off anyway so would rather get one we can at least see the settings.
Never ends at this joint....