According to a report written by The Tutor Pages, there was a large increase in requests for speech and elocution private tuition in 2011. Elocution was the subject with the highest number of enquiries per tutor (2nd highest was university level statistics). One of the elocution tutors cited in the report said she thought that the film The King's Speech was responsible for the growth in interest, with people thinking that if the king's serious speech impediment could be improved, then their lesser problems could be easily solved too.
All sorts of people want to change the way they speak - professionals, those in the media, even religious ministers. Some people want help with presentation skills and public speaking, but the commonest request is to help lose or soften a regional accent. The largest group of people with this request comes from Birmingham and the West Midlands. Many of these people feel they are being discriminated against in their quest for employment or promotion because of their accent. One enquirer said he'd been told he sounded 'thick' or 'like a thug'. However, most people don't want to lose their accent completely, just to tone it down and make themselves easier to understand.
Another large number of people approaching the tutorial agency are professionals from overseas who want to acquire a 'posh' accent. Joanna Lumley has been specifically mentioned as a role model (if you don't know what Joanna Lumley sounds like, click here).
Parents often approach The Tutor Pages with the aim of getting their teenage children to speak differently, and snobbism is often apparent in some of the quotes in the report. One Cheshire parent was horrified because his or her son had developed an 'irritating Scouse accent', another was horrified when his or her daughter moved from a private to a state school and the girl's accent changed.
Elocution lessons were quite common when I was a young child (I never had them, but at secondary school we had speech and drama classes, which were similar, I suppose). Elocution then fell out of favour for several decades, but seemingly it's in fashion again. Elocution was never considered a traditional academic subject, and even now I notice in the profiles of the tutors in The Tutor Pages report, that most have a singing or acting background, not one in phonetics and linguistics.
Here's the report from The Tutor Pages (in pdf format).