Is the use of the terms mummy and daddy to (informally) refer to one's parents particular to a specific socio-economic class or culture? How does this contrast with the terms mum/ mom, and dad?
I can definitely personally attest to it being used by lots of working class families, middle class families, families who would be "old money" by a lot of people's standards but don't have titles, and aristocratic families where there's still ownership of the estate (as opposed to "well yeah, I'm a baroness, but it doesn't mean anything and I don't use it", though that would cover my partner, and she grew up with that use too).
There's definitely a regional element to it. My using those terms wasn't remarkable for a working class kid where I grew up, while my partner's using them did mark her as middle class where she was.
In my experience, it's the working class that have the strongest regional variation; middle class kids have a mummy throughout the UK and Ireland bar a few with a mum and a small number having a mother, while working class kids may have a mummy, but may have a ma, mam, mammy or mum. Moms seem to be more common than they were, suggesting a US influence.
The tendency to use the terms into adulthood seems to have a similar relation to both class and region; some places marking one as working class, some places as middle class, and some places giving nothing away.
Kate Fox in Watching the English claims that using mummy as an adult is upper class while using it as a child is common to everyone.
I do not think it belongs to a special socio-economical class but I have heard it from children not adults in different socio-econimical classes with only one exception who is :
Evidence from the Corpus of Contemporary American English and the British National Corpus shows the following:
Mom is something like 30 times more popular in American English than either mummy or mum, and dad is almost three times as popular as daddy.
In British English, mum is over three times as popular as mummy, with mom not being very popular at all. Dad is also found three times more often than daddy.
These are crude results. Only a closer analysis with a considerable amount of research could determine the context in which each is found, and the extent to which they might be class indicators.
I grew up in Saginaw, Michigan. My Mum always called her parents Mum & Daddy. I as an adult (at present I am 56) I also have always call my parents Mum and Daddy. My daughter is nearly 24 and she still calls me Mummy and her father Daddy.
Born (1949) and raised in the Louisville KY area, spent most of my life in Southern Minnesota. My parents were "mommy" and "daddy" when we were younger, but about the age of 10 I decided (probably based in part on TV and in part on what my friends did) that they should be "mom" and "dad". My two (grown) sons have always called us "dad" (that's me) and "mom".
My grandparents on my father's side were "grandma" and "grandpa" when we kids were in earshot, and I'm thinking my father called them "mom" and "dad" otherwise (though maybe it was "pop"). My grandfather on my mother's side (in Mississippi) was "granddaddy" when we kids were around, and I'm thinking my mother called him "daddy" otherwise. (My grandmother on my mom's side died before I was born.)
I'm vaguely recalling that a childhood friend called his mother "mummy", but it was not the norm. I don't recall "ma" and "pa" being used at all (except on TV).