Perhaps a bit further than you would like, but Kingsbury Water Park would fit the bill and would make a full day out.
A combination of dedicated tarmac and gravel trails around several lakes - mostly level or thereabouts. The tarmac stretches are former main roads that have been bypassed for years and would be ideal for beginers. They are part of several circular trails that are well signposted.
Plenty of parking (tarmac or grass), toilets, cafe and bike hire. There is also a miniature steam railway that runs on Sundays and also Wednesdays in the school holidays.
Just outside the park and accessed from there via a canal towpath or road is the Dog and Doublet, a decent canal-side pub.
It's about 10 miles from where I live and is part of a circular route for me. It is a very pleasant place to ride and is open all year round.
https://countryparks.warwickshire.gov.u ... water-park
http://www.cyclenorthwarwickshire.org.u ... ng_leaflet