The 22-year-old said he had a “little chat” with the Princess Royal who said she was “incredibly proud” of him.
The double Olympic champion was the first British male swimmer in more than a century to win two golds at the same Games at Tokyo 2020 and has set his sights on more glory at the Commonwealth Games in Birmingham with Team England.
When asked how his day had been, he replied: “Not too bad. I was just saying I have had worse Thursdays. No, it was absolutely incredible.”
Looking ahead to the Commonwealth Games at the end of the month, he said: “I think we are going to come home with quite a lot of medals which will be exciting.”
He told the PA news agency that the success at the World Championships in Hungary has given him “so much confidence” going forward, adding: “I’m feeling really good. And it’s great knowing I’ve got results from just a few weeks ago.”
The 22-year-old said the crowds would make a “massive difference” as it “raises the whole team up”.
“They (the Games) are just up the road in Birmingham, (the) home crowd packed out, 5,000 people in that new aquatic centre,” he said.
He said it was a “privilege in itself” to see the inside of the palace.
Dean became a double Olympic champion with gold medals in the 4x200m freestyle relay and the 200m freestyle on his Games debut.
He was among an array of other sports stars collecting honours including fellow Olympic champion Dylan Fletcher-Scott, who was made an MBE for services to sailing.
Fletcher-Scott made Team GB sailing history at the Tokyo 2020 Olympics by winning the men’s 49er gold medal with Stuart Bithell.
Of his conversation with the Princess Royal, he told PA: “We talked about the future of sailing and inspiring the next generation of kids.”
He said that the next generation of sailing was “for everyone” and it was “faster and more fun”, and praised schemes that widened access to sailing.
“You’ve got the Andrew Simpson Foundation in Weymouth that are now opening up centres from around the world. And they take kids from any background, and you come and you sail for £5 and they teach you how to sail.”
He added: “For so many kids (it) is amazing because whatever your background, once you’re on the water, you have that freedom and you’re in control.”
Fletcher-Scott said his parents were ballet dancers but he “didn’t like ballet at all” so ended up getting into sailing.
“I ended up getting into it at 13 or 14 and started racing, and then just sort of progressed quite quickly through it all and here I am today.”