I used the OS Travel Map series, which I think have a little more detail than a road atlas, but I think that the road atlas + a compass would be sufficient.
I also had a more close up look at some of the towns/cities I was passing through on google maps to get an idea of how the roads worked.
I would think that with 4 weeks to do it you probably could make it up as you go along.
I only had 10 days and only really preplanned overnight stops and left the final route details to be decided either the night before or during the day.
With 4 weeks you should be able to avoid as many main roads as possible which will be great.
http://www.ukcampsite.co.uk/ gives a fairly comprehensive list of uk campsites which I have used previously to find sites in areas I do not know.
Have you made any decisions yet on route?
It would suggest investigating some of the National Cycle Routes - I followed large parts of routes 1,7 and 77 between John o Groats and Perth - these keep you on quiet roads and also some traffic free cycle paths. I spotted many other routes as I headed south but didn't know where they went so did not follow them.
The only issue with not preplanning/booking campsites in August may be getting somewhere to stay? It is school holiday time for Scotland and England.
Enjoy the planning, the only thing better is actually being out on the road.