MN is full of extremely wealthy, privileged people. By their definition, she was working class.
I'm a teacher and was told I am working class on HE once (in the context of someone telling me teachers don't need degrees , are all a bit dim , and students couldn't possibly actually aspire to a working class degree...)
I'd describe her background as 'ordinary' and she was always a 'grocer's daughter' when people wanted to talk about the beloved meritocracy. Boy, did she try and hide her origins (unlike poor old Keir who is trapped in perpetuity in the 'my father was a toolmaker' line).