I'd still classify Major as of working class roots. I suspect it depends on where you take the 'snapshot' of his life. He left school with a handful of O-levels, dad dead at 19 never went to university, was for a period genuinely unemployed and signing and flat shared with three families in Brixton. He pulled himself up as he entered his 30's of course and worked his way up in his career and in politics.
I see where you are coming from but I still do not consider his background to be working class. Where in the UK do you reside? I lived in Portsmouth for 14 years.