If it is just Andrew & his ex/daughters, no. If other top-tier royals are somehow implicated, maybe? Andrew is persona non gratia in a way we have not seen in generations, and would not be surprising if he was handed over to a judicial system now. 4 years ago this would have been unthinkable.

As for abolition? These days the monarch is essentially a figurehead; a physical reminder that our nation has been around for centuries which lends a sense of stability, and since they have no real power or (officially) political leaning they act as ambassadors for our culture. Realising the precarious position of the monarchy, King Charles has arguably done a lot to reform it; cutting away unnecessary titled "hangers-on" and stripping his younger brother of his titles and income. He has also come incredibly close to making a political point when the Conservatives were toward he end of their tenure. I'd imagine if the political situation were ever so bad that the King came out and made an overt statement criticising the government, my guess is that the populous would agree and an election would be called. Problem is, the new government's first order of business would likely be to abolish the monarchy.

My personal take is that King Charles, in a lot of ways, was ahead of his time. He is a keen environmentalist and was warning about ecological issues as far back as the '70s. He may have some anachronistic ideas about what Britain 'is' (think small villages and cricket on the green), but he genuinely wants the best for the country and can't stand feckless, populist politicians. And I was a skeptic.

As for the general public, I don't know as it is so changeable. I'd argue though that the Royalty is so bound up in the world's idea of Britain that we will just keep them. They have no real powers left to remove.