I have been informed from other sources that there may actually be reason to dislike Queen Elizabeth, for the record. Apparently, soon-to-be King Charles was naughty with some people of an inappropriate age, and the Queen covered it up to protect the family image.
One thing that doesn't always translate well outside the UK is that the monarchy isn't just this random family of private individuals, it's a massive quasi-corporate institution. I don't think the queen ever had that much control over her family's public image, they are all stage managed by an extremely expensive and well-oiled public relations team. While some members of the royal family do have a sort of independent public life and status, the queen's life was so integrated into her role I don't think she has done for a long time.
Charles (as far as we know) isn't accused of sexually abusing minors, it's his brother Prince Andrew. So far there has only been one case which was settled out of court, but given the circumstances that's essentially an admission of guilt. I have no doubt Andrew's bullshit was covered up for a long time, but I doubt we'll ever know how personally responsible the queen was.
Charles is, however, not particularly popular in the UK. He's less unpopular now than he used to be, mostly I suspect because the aforementioned PR team have gotten really good at keeping him out of the public eye, but he's a strange, uncharismatic little man who doesn't attract the same goodwill as his mother. His most notable public achievement was cheating on princess Diana with his current wife (our new queen consort).
Speaking of which, how good is Prince Charles health? And whose next in line for the throne?
I would imagine Charles has a decade or so left in him.
After that, we move on to Prince William. From the perspective of the PR machine behind the royal family, this would be a massive upgrade as William is considerably more well-liked by the public than Charles.
I mean, does England even give a shit about the royal family?
That's a complicated question, not least because England isn't really a nation. The short answer is yes, because our actual nation doesn't really have an organic reason for existing outside of the incestuous exploits of one family, and thus said family has made it their business to insulate itself as closely as possible into our sense of national identity and belonging.
Americans make their kids talk to a flag each day. We pretend a family of rotted, hemophiliac rich people are our friends.