I do feel your pain, I like to get out of the house even for my own sanity - our days fully at home are very limited as I just go a bit stir crazy
Did a quick search for library's in East Midlands (obviously it's a fairly big place but you can look up the ones nearest to you) and clicked on one at random (Kettering) which has things for under 5 year olds on Mondays, Tuesdays, Thursdays and Fridays. Granted the event itself (story time, play and learn etc) is only an hour but you can join a library for free, let the kids enjoy the hour with other kids, then pick a couple of books each and return the last ones.
Other things...
Castle? Talk about princesses and princes and knights etc, explore the castle, enjoy the grounds when the rain lets off and indoors when it doesn't. A lot of them have a nearby cafe or one inside the grounds
Pottery place? Some of them offer messy play which you can attend more regularly where a few kids go and just do painting etc which is quite fun. The pottery itself you probably wouldn't want to do more than about once a year (handprints or whatever) but worth checking if they do some other craft related activities for kiddos
Indoor mini golf? Some are very interactive, the young one can crawl about and the older one have a go and watch you playing it. I never did this with 2 of them just with my one child since he was about 3 or 3.5 and he loved giving it a go - recommend doing that in term time during school hours so there's only really younger kids who don't know what they're doing and no one's very annoyed if you just spend ages on one bit
I think they do under five sessions at trampoline centres and ice rinks too where parents are on with them - but I never tried those myself, probably under 1 is too young for those though the 3 year old would be okay
Museums - if it's interactive and interesting, like cars/trains/science/animals - not the history/artsy ones
Cinema - totally depends on the child but if it was a short childrens film mine would love that, I normally did M4J so that it's only £1.50 per child and same for adults - doesn't feel too painful if you leave before the end. Most cinemas have ride on cars just outside and an arcade somewhere within the area, plus some food outlets so it can burn through quite a bit of time just 'seeing a movie'... I swear mine only really asked for cinema because he wanted the arcade 🤣
Looking just now the national space centre Near East Midlands looks really cool! Mine would have loved something like that :)
Sometimes even just going to an area which has some interesting shops works for kids - e.g. aquarium that sells fish to go see them bobbing about, a shop you might actually want to go in, then pet shop to go see the rabbits etc, then another shop you want to go in, finished off with some food or whatever
National heritage sites have stately homes and sometimes a little park and somewhere to eat, park as and when the rain stops...