Do people in England have the reputation or "stereotype" for being very snooty or "snobby"?

The class thing still exists, in ALL parts of the UK. There is more to our country than just England! You have areas, considered snobby, in most parts. The South East and parts of the "Home Counties", around London, places like Gloucester etc... may be seen as such, from other parts, but a lot is simple stereotyping.