"Any name which rises too quickly in popularity can be considered not posh."
Don't think this is true. If you look at the telegraph birth announcements, the most popular names on there are very similar to the most popular names in the country in general. A lot of them are names which have risen in popularity very quickly - Florence, Kit, Scarlett, Archie, etc.
I'd say there are names which are popular across all classes - classic names and biblical names, for example. More recently as well, are names which used to be nicknames for classic names (you know, like Elsie, Daisy, Teddy, Eliza, etc - all of these used to be nicknames but are now popular among all classes.)
I've worked in public and private schools in the UK as well as abroad and haven't seen much difference in names except that there are some you wouldn't often find in private schools - names which are looked down on by people who consider themselves 'middle class' or above. Anything hyphenated, for example. Anything made up. Any of the names that rhyme with Aiden (Jayden, Hayden, Kayden etc). Lexi, Shannon, Demi, etc are some other examples.
If I were going to generalise, I'd say that from my teaching experience I've found that a lot of the more adventurously named children are from either working class or very posh families, and middle classes seem to stick with more traditional names.