Does it have to be walking distance, or are you ok with taking a car occasionally?
If walking only, then Chaterlerhault is probably at the limit from Bellshill depending on your fitness level, but I've camped there numerous times and as long as you keep a low profile and don't have a blazing fire, you're fine. It's getting to the 'busy' season down there though. I've done 8 overnights there this year so far, and a fortnight ago was the first time I've seen anyone else deep into the park at the time of night when you know they aren't 'day' visitors. Every summer is the same, but it does start getting deserted from September again.
If it's the car, then I can think of literally over a dozen within a 10 minute drive, but I'm very reluctant to post them on an open board. Nothing personal against you, but these are open boards and I've spent enough time picking up other peoples litter without advertising what might be new spots for the neds to trash.
If you're willing again to open up your radius to 25 miles travel, then you've got some of the best wild camp areas in the UK. I spent Friday night past in one, and with zero light pollution, it was, as always, a fantastic experience.
