Sissinghurst and Leeds Castle from Heathrow

The suggestion would be to just get to Canterbury and recover from your jet lag, then do either/both over the weekend.

Sissinghurst has no public transport- the bus #5 runs from Maidstone, but stops in the village and at the road end. It's a bit of a hike up from the road end.

For Leeds Castle use Bearsted or Hollingbourne stations on the Maidstone to Ashford line, then a short taxi ride or a 1/2 hour walk.

So maybe the alternative is to book a hotel in Maidstone, then go from Heathrow to Victoria, train to Maidstone East, drop your bags at your hotel, then do Leeds Castle.

Then on Saturday do Sissinghurst.

Sunday sightseeing in Canterbury (go from Maidstone by train), then train again to Canterbury on Monday to join the tour.

I'm pretty sure your proposal does not take into account the RADIAL nature of most trains in Europe. Your other stops are not "on the way" to Canterbury.

https://community.ricksteves.com/travel-forum/england/cotswolds-taxi-service-book-in-advance

https://www.tripadvisor.com/ShowUserReviews-g1593207-d215695-r709741731-Sissinghurst_Castle_Garden-Sissinghurst_Kent_England.html

there is another very nearly identical thread at https://community.ricksteves.com/travel-forum/england/sissinghurst-and-leeds-castle-from-gatwick-airport

The only difference is the airport. Since you can't use both at the same time, can you clarify which is the thread you want the answer to?

In fact the title says Heathrow and the question says Gatwick. Now I am really confused...

Since these are your only two posts it is hard to get the context.

I apologize for confusion. We’re flying into Gatwick and will be using public transportation to visit both Sissinghurst and Leeds Castle before going to Canterbury.

At least Gatwick is on the correct side of London.

But for both Sissinghurst and Leeds Castle you have to get to Maidstone first, and get a hotel in Maidstone- that journey takes about 2 hours.
Going across country it is 3 changes- at Redhill, Tonbridge and Paddock Wood- which is just awkward, especially if jet lagged.

I would choose to take a Thameslink train to London Bridge, then a South Eastern train to Maidstone East- only takes about 15 minutes longer. Then dump the bags at the hotel in Maidstone then the 1/2 hourly train as above, for Leeds Castle.

I don't think you would have time that day to do both, so then do Sissinghurst on the Saturday by bus, as above.

The key to all of this is that Maidstone is your base for the weekend.

Then on Monday train to Canterbury West changing at Ashford- total journey time around 75 minutes. Or there is a direct X3 bus at 22 minutes past each hour from Maidstone, takes 90 minutes. (no Sunday bus service)