Willow Grouse is the name I would prefer to use so I fully agree with Sanddancer but a problem arises when we want to say whether something is "correct" or not. Unfortunately there are a number of "authorities" and therefore a number of "correct versions". Howard & Moore, which I used to swear by, calls Lagopus lagopus Willow/Red Grouse with the latter name relating to Lagopus lagopus, scoticus. On the other hand Birds of the World, Recommended English Names, by Gill & Wright on behalf of the International Ornithological Congress, call Lagopus lagopus the Willow Ptarmigan. I still think it's a Grouse but, because I need to be consistent in the websites, I have elected to follow IOC.
The Black Grouse managed to keep its name in the IOC listing although its scientific name was changed from Tetrao tetrix in Collins to Lyrurus tetrix.
As Lapwing said " when you are up on the moors in the cold and the rain what is there to do other than look for bird poo and talk about birds names".