Which Beatle came from the poorest background?

The Everton area of Liverpool, 1960

All four Beatles had what Mark Lewisohn calls ‘unvarnished working-class roots in an industrial city that had seen better days.' 

Three grew up in adjacent areas in the south of Liverpool. Paul and George attended the same secondary school, while and John and George (briefly) went to the same primary school. 

Only one experienced absolute social deprivation and the perils of growing up in ghettoised poverty.

John

I moved in with my auntie who lived in the suburbs in a nice semi-detached place with a small garden and doctors and lawyers and that ilk living around... not the poor slummy kind of image that was projected in all the Beatles stories. John Lennon

Only during his earliest years did John experience anything approaching poverty.  In his final major interview with Playboy he describes in detail how his upbringing differed from his bandmates:

In the class system, it was about half a class higher than Paul, George and Ringo, who lived in government-subsidized housing. We owned our house and had a garden. They didn't have anything like that.

Paul's first impression of John's family was that they were 'very posh'. McCartney did not disapprove - the new social rules seemed like a logical extension of the ones his mother had taught him.

One of the immediate class signifiers that Paul noticed was John's use of 'Aunt Mimi' rather than the more familiar 'auntie'. Interestingly, John seems to have subconsciously corrected himself, using the word 'auntie' in the interview.

Paul and George

As Lennon points out, Paul and George (but not Ringo see below) grew up in social housing or what the British call council houses. These were allocated on a means-tested basis but neither family considered themselves poor. 

The immediate post-war years were characterised by generalised austerity rather than absolute poverty in the UK. In fact war-time rationing - which continued into the early 50s - had been introduced to ensure that everyone had access to substance level supplies of meat, vegetables etc. Subsidised clothing was similarly available through coupons.

There was also effectively full employment, with low paid work widely available. The new National Health Service had removed the fear of medical bills and there was widespread working class access to high quality education. John, Paul and George all went to high performing schools.

Ringo 

Tellingly, John implies Ringo came from a similar social background to George and Paul. In fact the then Richard Starkey grew up in Dingle, an area notorious across the city for its poverty and crime. 

The Starkey family had no access to the comparative luxuries offered by council houses. They paid ten shillings (£0.50p) a week to a private landlord for 10 Admiral Grove, a terraced house without a bathroom or indoor toilet.

The house where Ringo lived Copyright Pernille Eriksen — reprinted with permission — prints available

Ringo was the outlier of the future Beatles, experiencing a Dickensian childhood combining poverty, ill health, poor education and paternal abandonment. 

Ringo's Childhood: A Dickensian Chonicle of Misfortune 5 minute free read on Medium