I've been through this kind of situation with my own Mum, though for different reasons
She can be very volatile, unnreasonable openly favours my Brother & always has & enjoys trying to play us off against each other, regularly flies off & stops speaking to me, sometimes for years on end, mine can also be very manipulative & dishonest & plays the guilt trip to the hilt to get her own way, & in the past that has meant lying so that none of my family spoke to me on her say so...thankfully they know better these days.........& thankfully I don't live close to my Parents, so visits are not an issue,
But DD does get on great with both of her Grandparents, especially my Mum....as my Mum so kindly told me when she first met DD (over 1yr old, as she hadn't spoken to me then for 2 years ) "well at least you've given me in a Granddaughter, a real little girl, the kind you never were" [hmm} .........& I do see that DD s very "girly" & relates well to my Mum because she is too
so despite my Mum not currently speaking to me AGAIN I do encourage DD to ring her..........though DD has surprised me this time, & refused, turns out my Mum has upset her too.......
without my knowing, DD had told her Nana about a recent health scare I had......something I never bother to talk to my Mum about anymore as she has some pretty weird ideas on illness, so I've learned not to wind myself up.........but turns out DD is upset with her for whatever her reply was & in DDs words, "for my Nana not caring & asking if you are okay" when I later took the phone........DD is only 7 & I am playing down my own fall out with my Mum, as a "bit silly" ..........
so in my experience Kids can be surprisingly astute in these situations, but from earlier experience when my Mum kept deliberately sending DD parcels of things I had already said no too...including bras high heels, too old DVDs & Games.......PC games telly DD I would by her a computer to play them on, as we use MAC.& sweets my Mum knew she was allergic too....etc etc etc
The one VERY important bit of advice I will give.........is by all means keep up THEIR relationship with your Mum, if you know they are safe with her, then tthey have a right to that......
but DO make sure that they don't, as my own DD automatically presumed...that the pecking order is .....Grandmother is in charge....because she is YOUR Mother........simply because, as Mum, you are in charge of your own Kids.........until DD came out & told me this, I had HELL over the unsuitable gifts..........once I put DD straight, that is true for Children, but I am now a grown up, & HER Mum, so I'm in charge of DD....everything ran a LOT more smoothly
good luck