If you've not decoded on a site yet, you definitely need this review website. (It's like a specialist version of TripAdvisor):
https://www.ukcampsite.co.uk
Because that site caters for people camping under canvas and/or using their own touring caravans (as well as for people like yourself), you'll need to use the filters available to you (once you've got down to searching a specific county) to select 'Static to rent', as well as applying any other filters that might meet your needs.
Alternatively, since you've not got a car, you might like to start from this list of all sites that have direct access to public transport and then click on individual links to see which ones have static caravans to rent:
https://www.ukcampsite.co.uk/articles/view.asp?id=580
That website includes everything from sites that are little more than a few caravans in a field somewhere up to major caravan parks with loads of facilities (often with on-site entertainment too), so you really do need to study each entry (and its associated reviews) carefully to see exactly what's on offer.
Similarly, take care to check out the facilities within any particular caravan (or lodge/chalet) that you might be considering renting. Most modern units have their own WCs and showers but there are still a few more basic caravans offered for hire where you need to walk to a toilet block to go to the loo or take a shower.
Ideally you need to be looking for a site that's close enough to an interesting town for you to be able to walk there, take a bus or get a taxi (so that you'll have something to do during the day, perhaps seeking out a nice pub or bistro) but that's also got a clubhouse (with a bar, and perhaps some entertainment) to occupy your evenings.
If it's important to you, remember to check whethe there's a decent wifi signal any clubhouse there might be. (Read the reviews carefully. Also watch out for any references to there being hardly any mobile phone signal in the area).
Think also about meals. Do you intend to do most of your own catering? (If so, read the reviews carefully to see what any on-site shop might, or might not, be able to provide and/or think about where else you might be able to shop). Or are you hoping to eat out most of the time. (If so, read the reviews to check up on what any on-site catering might be like and/or do some research to find out where the nearest decent pubs might be).
Also check carefully as to exactly what's provided for you within the caravan. Most will come with bedding, etc, provided but a few might expect visitors to supply their own bedding (which could be a pain if you have to lug it half way across the country on public transport). Many caravans will have TV sets in them but some won't so, if you can't do without your regula fix of Corrie, that could be something else to watch out for.