I ran across this information while I was fantasy planning my Spring 2022 trip with my sister-in-law. I emailed the Royal Horticulture Society about next year's dates since a google search provided conflicting information. 2022 Chelsea Flower show is tentatively scheduled for May 24-28, a week later than this year's was scheduled.
This year's (2021) Chelsea Flower Show has been postponed until Sept 21-26. This will be the first time ever the show has been an autumn show. Since it was originally planned for May 18-26, it probably won't affect anyone's plans. But public general admission tickets for September are available on their website. All the information is on the RHS's website Chelsea Flower Show.