This year Kensington Palace has been more popular than ever due to their current exhibition, and the increased interest in all things Royal. Here's a little back story forum users may find of interest.
If you are visiting Kensington Palace or Gardens around lunchtime, I recommend the Zaha Hadid-designed Magazine restaurant at the Serpentine Galleries.
https://www.serpentinegalleries.org/visit/food-drink/
Good for a late afternoon beverage as well.
We went to the first morning of the Crown to Couture exhibit - it is gorgeous! It is definitely selling out and we saw multiple people get turned away before 10:30 am. Definitely book tickets in advance if you want to go.