Turn your question around; find a cycleway and then find a site on it....so I'd like to find a site near a cycle path away from traffic
Kielder Castle car park is where I stayed earlier this year. Loads of dedicated cycle paths there. Even some fantastic mountain bike trails too!There is a decent cycle track around Kielder Water . It has the attraction of being a Dark Sky area so the stars look good. Well, they do if it ever stops raining. Also, the track around Graham Water is a decent flat ride. I used to do a fair bit of towpath riding. Birmingham to Liverpool was the longest. It’s a bit narrow in places . A mate of mine met another cyclist coming the ther way and put his foot on the grass at the side of the canal. Unfortunately the grass was just some overhanging tufts. I honestly thought I was going to die laughing.
Stayed at Ashbourne Heights last month which is right on the Tissington Trail.. ideal as you have Tissington only a mile or so north (great cafe and lovely village) plus Ashbourne some 7 miles return right into town to the south... Ashbourne is one of our favourite townsA few sites off the tissington trail in the peaks and also the Monsal trail both very nice trails traffic free, callow top is one site or Ashbourne heights
Also there is a cycle track all the way around carsington reservoir and a site right next door