Which gardens are best with my itinerary

Check National Trust UK web site. Since you will be in UK for 16 days it may be worth your while to get a couples Touring Pass for 7 or 14 days. These were recently available from 47-57 pounds. Or you might become members to get more benefits. Seniors get a reduced price. The site is really functional and will give you listing of which properties are in the area that you are touring. You may have to buy a timed entry ticket for Stonehenge in advance; they are available in Salisbury. You seem to be zigg zagging a little bit on your tour. Going linearly it would be Windsor, Stay near Blenheim, Cotswolds, Salisbury, Stonehenge, Bath, Take freeway north(overnight Chester),North Wales, Liverpool, Lake District, Hadrians Wall, Durham Cathedral (Highly recommend), York, London. We did a similar trip. You may have to edit on the way. Don't worry about booking every stop if you have a car. You can stay in a country village like Hexham on Hadrians Wall. Britains major industry seems to be tourism so every grandma has a B & B.

Choose one of these in Yorkshire (if you have time)

https://www.rhs.org.uk/gardens/harlow-carr

https://www.nationaltrust.org.uk/fountains-abbey-and-studley-royal-water-garden

https://www.nationaltrust.org.uk/nunnington-hall

Day 14 - Hadrians' Wall, Durham Cathedral (if time) York

This will be one busy day. But these are three of my favorite things on your itinerary. So I would try to do them all. As Kathleen suggested, a stop for the night along Hadrian's Wall country might be a good idea. Hexham, as she suggested would be a good stop.

A great garden to visit is Sissinghurst, to the south of London, in the county of Kent. I don't know where you could fit it into your schedule, but it's fabulous.
https://www.nationaltrust.org.uk/sissinghurst-castle-garden
https://en.wikipedia.org/wiki/Sissinghurst_Castle_Garden
It could be done as a day trip by train once you get to London since by then you would have turned in your car.

I know it doesn't fit into your itinerary, in that you won't be going right by it. But it's so very good, I just wanted to mention it.

Sissinghurst looks amazing! There is never enough time on these trips. I'm wondering if there is any gardens to see between York and London as that is our most flexible day of the trip. I've already booked our hotels/B&B's so I can't change the schedule now.

Chatsworth! is a great stop between York and London. Actually, your tour looks just fine... I was just trying to save you mileage and some time by not having to retrace any steps. But Bath is a very pleasant place to stay. We used the National Trust Pass and saved a lot of money visiting the many sites included. Bon Voyage

Stourhead is not far from Bath. You might enjoy a stroll through that garden.

Day 3. You should be able too do Stourhead after Stonehenge & Salisbury. (It is a landscape garden rather than rows of flowers).https://www.nationaltrust.org.uk/stourhead

This one is very near Hidcote > http://www.kiftsgate.co.uk

You could detour to this one when going from The Cotswolds to NW Wales:> https://www.nationaltrust.org.uk/powis-castle-and-garden

Include this place in NW Wales >http://www.portmeirion-village.com and this one is on the Anglesey side of the Menai Straits:>https://www.nationaltrust.org.uk/plas-newydd-country-house-and-gardens

When going to Liverpool, be sure to take a look at Chester:> www.visitchester.com

Be sure to check opening times tally with your schedule.

Ellen, we took the train out of London to Staplehurst. Then took a taxi to Sissinghurst. We told the taxi driver what time we would like to go back to the station, so he came and got us at Sissinghurst and took us to the station later in the day. Here's their "How To Get Here" page from the website. You may prefer to go out another way. Bus maybe.
https://www.nationaltrust.org.uk/sissinghurst-castle-garden#How%20to%20get%20here

I see you have Day 16 through 21 in London on your schedule. You may want to think about doing the short trip out to Sissinghurst about the middle of your London days. Say Day 18 or 19. I find that several days of non-stop sightseeing in London is wonderful but tiring. Lots of sensory overload from the city, the museums, the traffic, the crowds. So my husband and I usually do a short day trip out of London by train right in the middle of our stay there on every trip. (The trains are very fast.) It can be very refreshing and relaxing.

Your itinerary looks good! Have a great trip!

I suggest Kew Gardens once you are back in London. Kew is a short boat ride on the river to the west of Big Ben and Parliament.

To add to Rebecca's suggestion of Kew Gardens, which I totally support, you will find nice gardens at Hampton Court Palace, and the Chelsea Physic Garden is well worth a visit too. The first too can be full days (pleasant days), Chelsea is a lot smaller.

No question. While in London, you must do a day trip to Sissinghurst. It is spectacular--one of my favorite gardens in the world.

Kew Gardens is so beautiful and restful. You can enjoy having lunch there.

So if I had to choose between Sissinghurst and Kew Gardens, which should I choose?

Tough choice, but I would say Sissinghurst.

The thing to consider is that Kew is actually within Greater London and easier to get to.

But better and more spectacular gardens.....Sissinghurst.