We've done some touring in our Speedster in the states, and have driven (rental cars) through a lot of rural UK, and in Europe, too. On our last trip, we drove from Southampton to Malvern (to visit the Morgan factory), then to Oxford, then down to the coast west of Weymouth, and back to Gatwick.
I found UK rural roads generally better maintained, if narrower, than similar roads here in California.
We carry a spare in the Speedster, but in seven years have never had to use it. It's a narrow 135/65 tire on a 4.5" rim that just fits. We don't use a luggage rack.
For us, the key is a full tonneau cover. Fully deployed, it will hide your luggage while the car is parked. Half deployed, it covers your luggage while driving, and being soft can accommodate taller bags. We find we can get two large backpacks or duffel bags behind the seats plus a third smaller bag and some loose sweaters and jackets, with a spare pair of shoes or two on the rear floor. This is enough for a few weeks of travel if you pack smart.
The spare and some tools live in the boot.
I guess it helps that we did a lot of bicycle touring through Europe for many years so got used to packing light, but it can be done.
One thing bicycle touring taught us was not to travel in the rain. Caught in the rain, we'd just hole up where we were for a day or two until better weather returned. This is probably good advice for Speedster touring, too!
Cheers!