A few musings in no particular order assuming based on wording of OP this is a scenario where a Charles dies at the same time in roughly the same circumstances as Diana did OTL.
Succession and Prince Andrew
As others have noted succession is actually pretty secure William becomes heir followed by Harry until William has children, Prince Andrew may become a little bit more prominent filling in as one of the most senior adults in the meantime, but I don't think this terribly changes his long-term role in the family and feels unlikely to butterfly away his numerous flaws so likely in similar situation to OTL
Diana
The big awkward question here is Diana, on one side I expect she'd lose favour with the press relatively quickly (leaving side how in general media/press tends to turn on women as they age anyway you also have the issue of Charles will automatically be seen in a more positive light following his death, and she lacks the press contacts/resources of the Royal family). However, by all records she had an excellent relationship with William and Harry, and with Charles out the picture that's going to become even more important so I'd expect a very tense relationship between Diana and the Palace around the role of particularly William and to a lesser extent Harry until the two are comfortably into adulthood. Having said that William particularly has generally seemed relatively comfortable, with his duty/role so while likely more strained then over to OTL I don't expect it's going to lead to radical a departure, you likely see a substantial rehabilitation of Diana within Royal circles as the Queen starts to withdraw and William gains more influence and power, sort of mirroring OTL increased presence of Camilla as Charles gets more authority (William would think his mother was hard done by, and her decreased personal media presence makes what the Royal family can offer more valuable). I imagine by today young people would be very confused if people pointed out that Princess Diana, generally seen as the Prince of Wales support act/often pictured with her grandchildren used to be seen as a glamorous/controversial figure.
The transition to William
assuming hordes of butterflies don't change the Queen's life expectancy the contrast between a young heir with a young family and the monarch feels much more stark then the Queen and Charles OTL I expect they'd be a bigger pressure to hand over more of a public role sooner, I doubt this would lead to abdication given the Queen's personal views on the matter but it wouldn't surprise me if we saw what we have already had happening recently but maybe 3 to 4 years earlier, with the Queen withdrawing from more elements of public life that aren't core constitutional and William generally becoming the public face of the firm.
Harry
Harry is likely under significantly more oversight in his youth as he will be next in line to William and generally have more of a public role, plus he seems to been practically impacted more by Diana's death OTL, and I expect would significantly benefit from ongoing presence. So likely less of a period of wild years then OTL, I also expect to get some actively encouraged to marry off a bit earlier (seen as a more reliable part of the firm), either way probably slightly better integrated with the Royal family although likely still with some tensions.
Charles's legacy
I'd expect Charles is heavily lionised after his death with the most generous possible interpretations of his different interests and priorities in particular I'd expect some sort of Prince Charles foundation set up by either William or Harry (if not both of them) centred around the environment/conservation issues becoming a very big deal by the 2010s, with Charles increasingly cast as a prophet on green issues who died before his time/before these issues became mainstream.
The aftermath
When Diana died she was a private citizen's, Charles on the other hand is heir to the throne an important part of the UK constitution, I'd expect the British government to demand an immediate criminal investigation with serious teeth and would be amazed if (almost irrespective of whether they were really at fault), some of the journalists involved did not face serious jail time. I agree with others have said this would kickstart greater press/paparazzi regulation earlier and again likely with substantial teeth. I'd also assume it would lead to a slightly more cautious monarchy on the roads possibly something more like US style motorcades.