Hi,
Check out
this site, it may contain the info you need.
I think you very well could see a great amount of the UK in just one month, but it can vary from traveler to traveler. Everyone's different in how they prefer to "sight see" so for some one month might not be enough but for others, one month might be plenty! For me personally, one month would be plenty.
Good luck and let us know how the trip goes!