This is not true, at least in practice.

The early Beatles, I think you'll agree, are rock and roll. They played songs live over BBC radio (sometimes recorded earlier, and as captured on the Live At The BBC collections released in 1994 and 2013), on several dates in 1962 and 1963 in particular: 7th March 1962, 11th June 1962, 16th January 1963, 22nd January 1963, 6th March 1963, 16th March 1963, 3rd April 1963, 19th April 1963, 1st April 1963, 4th April 1963, 21st May 1963, 24th May 1963, 11th June 1963, 18th June 1963, 23rd June 1963, 25th June 1963, 29th June 1963, 4th July 1963....honestly, I've gotten bored of listing dates, but they stretch on through to 1966, basically, because The Beatles were very very popular indeed in the UK in 1963, and the BBC were very happy to play very very popular music. The Beatles, in this regard, were particularly popular, and the BBC had them visiting the studios with unusual regularity, but other bands you'd probably call rock & roll - Gerry and the Pacemakers, the Kinks, the Rolling Stones - also played live on air on the BBC in this time period.

However, it is good to remember that the BBC in the late 1950s and early 1960s was a much more monocultural setup than we would be used to as people who could pick a 'radio station' designed by a streaming service algorithm. What did change in 1967 was the introduction of BBC Radio One, which was a radio station devoted to pop music for young people - which in 1967 was going to be influenced by rock & roll. Before 1967, the BBC Light Programme had been the main home on the BBC for pop music, and by extension rock & roll. However, the BBC Light Programme was not a rock & roll station, as it was meant to have broad appeal, and rock & roll, to the ears of BBC policymakers, appealed to the young.

Thus, the BBC Light Programme had specialty shows, often at night, to cater to younger listeners. From the 1950s, these were increasingly likely to focus on rock & roll, and these specialty shows - e.g., 'Saturday Club', which ran from 1957, and one of the shows the Beatles appeared on live - were the most likely place to hear rock & roll on the BBC. The BBC also was conscious of the musicians' union, which in practice meant that, where possible, the BBC would play versions of songs live, often by a jazz-oriented studio orchestra; Elvis Costello's father Ross McManus was one of a set of studio musicians who might be heard on the radio doing a (slightly too jazzy for the tastes of the young) version of a popular rock & roll song rather than playing the original. The Beatles promptly joined the union when they got the opportunity to play on the BBC.

So when a teenager heard an Elvis song on the radio, it was probably more of a rarity than you would expect - it would not have been like hearing a song from a modern high-rotation commercial radio playlist, but instead would have been something of a rarity.

The BBC had something of a stranglehold over the British airwaves officially, but in reality, they also faced English language competition aimed at British audiences but based outside of the United Kingdom. Radio Luxembourg began in the 1930s, and had a commercial orientation (compared to the BBC which didn't have ads). As such, they did not have the same agreements with the musicians' union that the BBC had, and they would take payola from record companies, and this meant that - considering the popularity of rock & roll - they were very happy to play rock and roll records as part of their programming. In 1964, Radio Luxembourg were joined by another 'pirate radio' station, Radio Caroline, which had a more modern pop emphasis (and which was an influence on the 2009 movie The Boat That Rocked (UK)/Pirate Radio (US).

Additionally, British teenagers had other options other than the radio for discovering new music. Firstly, there was the medium of film: the 1955 film Blackboard Jungle was in many ways the ground zero of rock & roll, because it was where many white teenagers, both in the US and in Britain, first heard rock & roll in the shape of Bill Haley and the Comets' 'Rock Around The Clock'. Elvis, of course, was a film star as well as a rock & roller; across 1956-1958, he appeared in four films, Love Me Tender, Loving You, Jailhouse Rock, and Kid Creole. Additionally, the period had 'jukebox movies' such as The Girl Can't Help It, which in practice was really a set of music videos for different acts such as Little Richard or Eddie Cochran strung together with a putative storyline. In terms of getting visuals for rock & roll acts, Top Of The Pops broadcast on the BBC from 1964, and - considering that the top of the pop charts were full of rock & roll bands in 1964 - was going to regularly feature rock & roll.

Finally, record stores like NEMS in Liverpool - run by Brian Epstein before he got a bit distracted managing some rock & roll bands - were another way people could listen to music they had heard about: such record stores had listening booths where people could listen to 7" singles over headphones before they bought them. The young Beatles apparently exploited NEMS listening booths in order to hear various rock & roll records imported from the US, without spending quite as much money on those records as NEMS staff would ideally have liked.