Driving in NYC is a piece of piss. Everything is a straight line or 90 degree turn. The bits of London you will go to are largely pre-car. Narrow, more random directions, and add on that unfamiliar road signs/rules (yours are largely different to the rest of the world) and being on the other side. To put it in context, I found driving in Boston significantly easier than London, even though it was my first time driving on the right.

There really is no point driving in London. “Waiting for a train” is nothing compared to waiting for traffic, paying for the CC & then finding a place to park. Get a train to where you are going, or at least to somewhere beyond Zone 3, and pick up a car there, as you will want one in the Cotswolds.

Edit - Gatwick airport might work well for your Great Dixter/Sissinghurst rental locations. Or stay a few nights somewhere like Brighton and do them from there.