GPS necessary with car rental in places like Rye, Bath, Cotswolds, Windsor?

Yes, your smart phone will work as a GPS. in the UK if it works that way at home. You don't need any mobile service to use the phone as a GPS, but you will need to make sure you download the maps ahead of time. Google Maps for example has an "offline" navigation mode (for driving only) where you download an area map. (The larger the area, the more space it will take on your phone.)

I suggest you try it "offline" at home first - unless you are sure you will have working mobile data on your phone in the UK (which also gives you things like traffic info in the navigation). But "offline" Google Maps works just fine in my experience, too. Download your home area's "offline" map into your phone in Google Maps, then put the phone into airplane mode and use it to navigate somewhere by car, to see how it works. It will the same way in the UK, though you'll need to download that area into your phone. (Maps expire/must be updated every 30 days.)

Just make sure you get a USB power adapter for the car's "cigarette lighter" power port ($1 at Dollar Tree in the US). GPS navigation drains a phone's battery pretty quickly. Some cars now have USB charging ports but you'll at least need the USB charging cable to plug into the USB port. And of course you may want a windshield suction cup.

We have enjoyed eight road trips in the British Isles, exploring all sorts of little country roads without needing a GPS. We use a Michelin map for planning and a road atlas for navigating. Both are at a scale of about 4 miles to the inch. These work very well for driving in the country and smaller towns. A GPS or Google maps would be useful for navigating in larger towns such as Salisbury, especially for locating a hotel.

Whatever you use for GPS (or as the Brits call it, satnav), just be sure that the driver does not touch it at all while driving. This includes while stopped in traffic or at a light. This came up in another discussion. They've gotten serious as all heck about distracted driving ing the UK - and good for them! You'll need to be sure you get your destination info entered and route chosen before you start the car, and your passenger will have to be co-pilot and handle the device en route. Or if you're the driver, you'll have to pull off the road and turn off the car if you want to fiddle with the thing yourself.

Here's an article about it: https://www.independent.co.uk/life-style/satnav-mobile-phone-drivers-illegal-police-fine-200-npc-bank-holiday-roads-gps-a7912191.html

We did a 4 week drive tour of England and Wales and stayed away from London the entire time, except picking up the rental car at Heathrow.

We did Bath, Chipping Campden and many other places. As Americans normally driving on the right, driving on the left take a lot of concentration, especially when turning, going around the many roundabouts, etc. Still, we managed quite well.

In my opinion, a nav system, gps or mobile phone with google maps is essential. I would go with the nav system in the vehicle as a first choice. Still, our Europcar rental's nav system was not always helpful in finding our B&B or small hotel. Sorry, but in a city of any size, you need some kind of NAV system or GPS.

Here is the story of our trip.
https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139