We have done many cruises - 2 were with my entire family, including 4 kids. Oldest was 11 and youngest was 2.
We didn't go on Disney. It interests me. I've heard the rooms are the biggest, but it was $$.
We did one cruise on Carnival - sorry to those who like this line but my family didn't enjoy it as much. We all had fun together and there was plenty for all ages, but the food, decor, rooms, staff were just not as good as on other lines.
Our favourite cruise with the whole family (10 total, 4 kids) was on Celebrity. What a fantastic cruise line! The food, staff, rooms, decor, activities, etc were all fantastic. We also loved the fact that it wasn't a super massive ship. The massive ships (like RC) are cool because they have a lot going on, but lines (for food, elevators, getting on/off ship, etc) are incredibly long and there was a ton of standing / waiting / crowds. Even getting from one side of the ship to the other took forever. For the sake of your elderly grandmother who has to get around I'd pick a smaller ship with fewer people. Celebrity still had a ton to offer everyone with many many activities. There was no climbing wall, ice rink, etc but we've been on RC ships with things like that and the other ships are just as nice.
Truly - our celebrity ship had almost 3500 people on board while the RC had an extra couple of thousand. That is a lot of people!
Anyway, we loved celebrity, really liked RC except for the size, and didn't love Carnival, although we all had fun and it has more to do with the ports and the people you're with versus the ship.
I would maybe try Disney but my girls are already 7 and 10. I'd probably prefer Disneyland to a ship catered to them, especially since their cruises are more expensive.
Have fun deciding. Planning is part of the fun!
Eta: on the carnival cruise my little dd was 2. She had to be toilet trained to go in the pool. Since she was starting 3 yr old preschool in the fall anyway (we went at the start of summer) and needed to be trained for them we decided to train her. Otherwise she couldn't have gone in the pool (if I followed the rules). No swim diapers were allowed in the pool.
and they are my go to also. But check cruisecritic.com for extensive information and I agree using a Travel Agent - ours was so much help. 
and they are my go to also. But check cruisecritic.com for extensive information and I agree using a Travel Agent - ours was so much help. 
It was way too much for my mom especially navigating the airports, waiting in the insanely long line to board the ship, and even some of the bus tour shore excursions had more walking than we thought. My mom didn't normally use a wheelchair at the time but it probably would have been helpful to have one.