So, DP is putting his foot down, and saying he does not want my mother in the house, and that if she "turns up unexpected" he will walk out - albeit probably only for a while. He doesn't want to see her, or have anything to do with her, and preferably doesn't want our DC to either. (They are 2 and 4 months respectively, btw.)
My mother, obviously does have previous on turning up unexpectedly, and of telling me how to live my life, as well as, on a number of occasions, to leave DP. (some of these may have been me reading too much into her comments).
DP also has depression and anxiety problems, and really does not cope with stress or the anticipation thereof. He's going through a bad patch at the moment, and has admitted to me that he feels low enough to have contemplated suicide, though he would never attempt it because he is terrified of not being remembered by the DC. 
Obviously I can't fit the whole story in one post, but there's also an element of my mum being a moderately rigt wing spiritual type (who has made some remarks which come across as quite bigoted in her time), and DP being a liberal atheist. Generally, they are probably never going to get along.
But is DP overreacting by saying he doesn't want her in the house? She lives the other side of the country to us, and I see her once, maybe twice a year for a few hours each time, as when she comes up she stays with my nana and tries to fit in visits to other people she knows here too.
He says I need to put my foot down and say something to her about her behaviour (which has been enough to make me cry before now), and I say that he has no right to deny our DC access, as she is never unpleasant around them, and I hardly ever see her anyway - not to mention if I cut her off, I will make it really awkward to see my younger brother and sister who still live at home, and who I love to pieces. (And who she will then spend forever moaning about me at). I'd rather just keep the peace, and then blow up if she ever says anything again, which she hasn't for a feew weeks. But that's not enough for DP, and I don't want a lazing row with him right now, as he's quite emotionally fragile.