I live in Leam.
The loop mentioned above at Ufton is a good shout. That section I normally do as part of a bigger loop that starts in Leamington and goes through Ufton, Harbury, Hunningham etc.
To be honest though, you’d need an OS map.
There are nice loops to be had from Leamington/Warwick out to Kenilworth castle and back with a number of different bridleway/cheeky options but again, a map is needed to make the most of it/not get lost.
Viewranger app I think has a few routes on it if you do that and can make a search.
I record my rides on Strava which I’m more than happy to share with you if someone can tell me how I do that 😀
It’s not the Peak or owt but it’s a nice area to bimble about in. Warwickshire has some beautiful scenery.