It is difficult post 1900 given the way the UK's monarchy had already evolved by that point. Even the biggest crises it has faced in that time (abdication crisis and Princess Diana) were unlikely to escalate to the point when the institution itself was abolished. The best way of doing it to my mind would be to have the UK have some kind of revolution or somehow become a dictatorship. Perhaps a loss in WW1 sets it down the path to becoming Communist or Fascist later on during the depression. A bit less plausibly, you could have things get far worse in the 1970s/1980s, resulting in a military dictatorship. In a Communist revolution, of course, the royal family would be overthrown, whereas in a fascist or military dictatorship the monarch could cooperate with the regime to the extent that there position could be untenable whenever democracy resumes. I can see no reason why that government wouldn't take the form of a parliamentary democracy, with a relatively toothless President replacing the monarch.
Of course, how the monarchy is remembered very much depends on the circumstances in which it went and what came after. If they were overthrown by a revolution that produced a regime that was eventually itself overthrown, they might be remembered sympathetically,especially if they met a bloody end, but at the same time seen as a bit of an anachronism. If they were complicit in a discredited regime, it would be very much seen as a case of 'good riddance'.
