
Just to add to what
@gregjrichards explained the streets & roads in London were packed, it was a security nightmare.
Althorp is also quite some distance from London which getting to by car means travelling on the motorway. If you don't know anything about our motorways, any incidents/hold ups would pretty much have a knock-on effect causing chaos throughout the whole country

(+ the whole security thing as well) Even so they still did have issues for concern with people along, above, the entire route throwing flowers etc
As for the Prince's walking with the casket

this article gives a good insight into decisions they faced & a lot of what went on that week.
The Complicated Truth About the Royal Family's Reaction to Princess Diana's Death
...
Neither brother initially wanted to walk behind the casket in the funeral procession to Westminster Abbey, but their grandfather Prince Philip—who, like his wife, also had a complicated relationship with Diana when she was alive—encouraged it.
"If you don't walk, you may regret it later," he told William,
"I think you should do it. If I walk, will you walk with me?"
William and Harry solemnly joined the procession as it passed St. James' Palace, making for one of the most memorable news images of all time.
"I don't think any child should be asked to do that under any circumstances. I don't think it would happen today," Prince Harry told Newsweek last year. But he also said in Diana, 7 Days that he was "glad" to have done it, whether it was right or wrong.
Just last year, Charles Spencer told BBC Radio 4 that he had objected strongly to the idea of his nephews taking that long, public walk, calling it a "very bizarre and cruel" thing to be asked to do. "Eventually I was lied to and told they wanted to do it, which of course they didn't but I didn't realize that," he said.
"It was truly horrifying, actually," he further recalled. "We would walk a hundred yards and hear people sobbing and then walk round a corner and somebody wailing and shouting out messages of love to Diana or William and Harry, and it was a very, very tricky time."
The Complicated Truth About the Royal Family's Reaction to Princess Diana's Death | E! News UK