Private shuttle service in England?

They are called Taxis!

Sometimes they're called Uber.

A shuttle is for a group of people or a shared service. What you want is to hire a care with driver. Most of the companies that will pick you up from the airport will also arrange to take you where you want to go. You have to contact them.

Where do you want to go?

Frank, thanks for the civil reply. I'm exposing myself to a potential snarkfest on behalf of friends who want to visit relatively hard-to-reach places in England such as parts of the Cotswolds and Hadrian's Wall, but are queasy about driving there. I was wondering if there was some equivalent in England to a company like GoOpti in Czechia (for example, we've used others) that uses mini-vans for people to take collectively, if enough sign up for a given destination, or otherwise privately, at a higher rate. One (other?) difference from taxis or Uber (not that I was paying close attention) seems to be that what I am calling "shuttle" drivers will drive from point A to B, and then overnight at B, awaiting further trips, rather than necessarily heading back to a home base after a fare; as though the company has "nodes" over a large area. I suppose our friends can always arrange with local tour companies that specialize in places where public transport is inconvenient.

I haven't read anything on this forum to suggest that there's such a company operating in England, but maybe there is. Your friends could check out the mydaytrip.com website for preliminary information. I believe this is a third-party company that just connects travelers with local transportation providers who provide private point-to-point transportation with (potentially) stops for sightseeing along the way. Those same companies could probably provide the service your friends want.

The Cotswolds area, while not tiny, is small enough that I don't think the driver would need to overnight away from home. Are your friends aware that there are small-group (van) tours of the Cotswolds operating from Bath and from Moreton-in-Marsh?

How many people and where do you want to go?

Are you completely unwilling to take trains to some points and then meet up with a driver there?

Where do you want to go in England?

And by saying point to point are you saying from Westminster Abbey to Hampton Court? From Oxford to Brighton? From London to Bourton on The Water?

"shuttle" drivers will drive from point A to B, and then overnight at B, awaiting further trips, rather than necessarily heading back to a home base after a fare; as though the company has "nodes" over a large area.

I've never heard of such a thing here.

Most people would use a local taxi.

Thank you all. It looks like the best way to go for them is standard conveyances such as Uber, taxis or trains, to be supplemented by local tour companies for areas like Cornwall, the Cotswolds, or Hadrian's Wall (I'm not sure where all they want to go, possibly at this point they are not either). Of course, a RS tour might be a really good option for them, if any were still open for this year.

acraven, thank you also for the link.
Edit: It looks a lot like what I was looking for.