I'm pretty sure that your grandmother was my mum's best friend from childhood; I knew her as Auntie Glad. She was smiley and always had her lippy on
I think she was tall like my mum but then all adults are giants when you're a little kid!
They were pals throughout their days at Gray St school which in their day was primary and and senior school. The school moved to new premises and was renamed Thomas Gray Primary many years ago but the old building stood until very recently. You'll find pics of it's demolition on this site somewhere - Bob Hammo and JD Dry may help with that.
Mum and Auntie Glad saw much less of each other after the mid/late 50s when the Pearsons moved to Netherton (Hereford Drive?); no one had cars then. As i wasn't born until 1950 I don't recall much but I do remember her living in Warton St and mum going to visit her after she moved. Although I don't remember a son named Alan I do seem to recall a son named Geoffrey? and was it two daughters? Don't remember ever meeting them as they were a fair bit older than me.
I remember Jimmy Pearson more as he had an allotment in Litherland near my dad's so I occasionally saw him into the 1970s. Think he also worked at the English Electric like my dad? Seem to recall a spot of rivalry over the flower shows there
As to the shop .... the one on the corner of Warton St and Warton Terrace was always the Parker's shop in my childhood days so perhaps it was your family's prior to that. Again Bob Hammo would be your best help with pics of that area as well as perhaps looking up the street in his directories.