I think it may have been different for each grandparent.
Obviously with their limited screen time in the first film, probably because they wanted to focus on Littlefoot's close relationship with his mother before ripping our hearts to shreds with her untimely demise, the relationship between him and his grandparents at this time is open to a lot of speculation.
Grandparents do typically spoil their grandkids whilst the parents are in charge of caregiving as well as discipline. But most of the time the grandkids don't live with the grandparents. Littlefoot has always had Grandma and Grandpa Longneck in his presence full time, so I'd say they were always close, and they probably did step in with helping Mama out if she needed them to. It's heavily implied that he's the last of their legacy, after all, so I'd say they were always pretty protective of him, too. But I'd also say the bond was different with each grandparent.
Grandpa was
always Littlefoot's father figure. Bron may be in the picture now, but I'd say if Littlefoot had to get some advice about something, his grandpa would be the first port of call. I'd imagine that, while he left most of the actual caregiving to Mama, he would tell Littlefoot bedtime stories and give him helpful advice, much like he does in the sequels. I don't think much would have changed between the two, save for Grandpa doing more of the caregiving, after Mama Longneck's death.
The biggest shift in dynamic would be between Littlefoot and Grandma, since she didn't start off being his mother figure but had to become it later on. I can imagine Grandma being more laid back originally, perhaps stealing Littlefoot for grandma cuddles, singing him lullabies, and, if the environment of the first film allowed it, sneaking him extra special food behind her daughter's back. But when Mama passed away, she had to step into her daughter's shoes and probably took over most of what Mama was doing. If you look at the sequels, she's the one setting the bedtime routine, sorting out his baths, etc. I wonder if Grandma and Littlefoot struggled with this shift, both having to deal with their grief of losing Mama Longneck whilst Grandma had to step in to do most of what Mama had done before.
Both grandparents would have had to adjust since Mama Longneck was probably doing most of the parenting work, but I'd say Grandma faced the bigger change with her relationship with Littlefoot. Their love never changed, but how they expressed it had to when they became Littlefoot's primary caretakers. That's my take on it anyway.

Honestly, I feel like the first film should've delved a little more into Littlefoot's grandparents. It would've been very interesting to see how they were like during that time, and what their interactions with their daughter were like.
Agreed. I think it would have made the ending more rewarding too, if we had seen Littlefoot interacting with his grandparents and it showed just how strong their bond was.