As a rule of thumb, if the mother of the children is very close to her own mother then I think it's likely, that the maternal side will take precedence. Of course everyone has a different set of circumstances, Speaking as a paternal grandmother, I feel my role is secondary, I'm not saying this bitterly, more as a matter of fact. The other side's input with time is greater, for instance gd will stay with them for weeks at a time in school holidays and she will be taken on holiday as well. When our own children were young we dreamed about adult holidays and breaks and whilst we might do something such as an occasional Centre Parcs trip, we don't want to do child oriented holidays all over again, we have waited a long time to be free of all that. I see my grandchildren regularly on a weekly basis and have our gd overnight on occasions and hope that will continue. I provide support with reading, swimming, outings to the cinema, trips to the park, I buy them clothes and shoes and make other financial contributions. I know my role and influence is secondary to the maternal grandmother, but my attitude is to go with the flow Life's too short to get into battles over who gets the children at Christmas etc. My ethos is to be the grandparent I want to be, I take the view that everyone brings something different to the table and those things wont always be the same as the other side, but will no doubt enhance child/rens lives and that's what matters in the long run.
My heart will always go out to those who have been cut out of grandchildren's lives.