I was born in the sixties. Yes Emily and similar names like Lucy, Charlotte, Olivia, Alice, Chloe, Isabel and Sophie were at the time seen as posh girl names, usually if you knew a girl with one of those names she went to private school.
Remember Bagpuss on TV in the 70s? (well, probably you don't..) 'There lived a little girl, and her name was Emily ...' I think it started becoming fashionable in the 1970s, slowly, and really got going in the 80s.
My classmates at state school had names like Claire, Sarah, Amanda, Louise, Karen, Tracey, Alison. And more traditionally, Jane, Anne, Catherine, Susan.