An interesting study from the BTO and Natural England suggests that grey squirrels aren't to blame for woodland bird declines. Nice to know, but not a great surprise really given that many (not all of course) woodland species are declining across Europe and there aren't grey squirrels elsewhere in Europe (apart from part of northern Italy).
It would be possible that woodland bird species decline for different reasons in different parts of Europe but that would make life rather tricky.
But whether or not grey squirrels cause problems for woodland birds - and you would have to ask whether red squirrels would have caused more or fewer problems if they were still around - the problem that is firmly laid at the door of the introduced non-native grey squirrel is the demise of our native red squirrel.
Yes Susan definitely causing decline in Reds as they are immune to some disease but pass it to the Reds who have no immunity and it almost always kills them.