I'm thinking about trying to organise a biking trip to the UK
WHY???????
Unless we get a miracle and have a good summer it will be cold, wet and muddy.
If you are riding on the roads, they are badly kept, too busy and the cars will treat you like you don't exist.
If you are riding off road, unfortunately, the majority of rights of way in the United Kingdom are footpaths and you can't ride on those (legally). There are some great forest trails and a few specific riding centres, but to do general off roading usually takes quite a bit of local knowledge as you can't just head off into the hills.
If you really want to ride in the UK, you could try something like the C2C (Coast to Coast) LINKY. There are off road sections within the ride and you will hopefully get to experience some stunning countryside with not as much traffic.
You might also want to have a look at SUSTRANS. They have lots of cycle routes, although mainly on road, but they would allow you to cycle between some of the better off road areas on less congested routes.