Last night, BBC One shifted its schedules to broadcast a film about the making of the “final” Beatles single, Now and Then. It was brief and rather moving, but it offered a tactfully bowdlerised version of events, understandably stepping around the parts of the story that might cause anyone to regard Now and Then with a wary eye. It talked about the surviving Beatles’ initial attempts to work up John Lennon’s late 70s demos in the mid-90s, but didn’t mention the slightly muted response the completed versions of Free as a Bird and Real Love received. It was the height of Britpop, the Beatles’ stock higher – and their influence on current music more obvious – than at any point since their split. And yet Free as a Bird – clearly released with the intention of bagging the Christmas No 1 spot, as the Beatles regularly did in the 60s – couldn’t dislodge Michael Jackson’s Earth Song from the top: by its second week in the charts, it was being outsold not just by Jackson, but Boyzone’s cover of Cat Stevens’ Father and Son.

Real Love, meanwhile, managed a couple of weeks in the Top 10 before disappearing (by week two, Boyzone were outselling that as well). Perhaps it was stymied by Radio 1’s disinclination to play it, which led Paul McCartney to pen an angry article in the Daily Mirror, decrying the station’s “kindergarten kings”: whatever your take on the issue, there was something a bit unedifying about the Beatles’ return ending with Macca fulminating about Radio 1’s ageism à la Status Quo. Moreover, the overdubbed recordings had an eerie, uncanny valley quality. Everyone involved had clearly done their best with the technology available but there was no getting around the fact that Lennon’s voice sounded ghostly.

The new film discussed technical issues hampering the surviving Beatles’ intention to rework Now and Then in the mid-90s as well: there were meant to be three “new” Beatles songs, one for each volume of the Anthology compilations, but the sessions for this song were abandoned as Lennon’s vocals and piano couldn’t be separated for the new mix. This was a slightly different version of events to the one given by McCartney a decade ago. Then, he claimed the late George Harrison – always the most unbiddable ex-Beatle – had singlehandedly drawn the sessions to a close by describing Now and Then as “fucking rubbish”. (“But it’s John!” McCartney had apparently protested, to no avail: “This is fucking rubbish,” Harrison countered.) Indeed, Harrison seemed unsure about the whole idea of reworking Lennon’s material. “I hope someone does this to all my crap demos after I’m dead – turn them into hit songs,” he subsequently remarked, which perhaps wasn’t the promotional boost for the new songs Apple was after.

Listening to Now and Then, it’s hard to see what Harrison’s objection was in purely musical terms. A moody, reflective piano ballad, it’s clearly never going to supplant Strawberry Fields Forever or A Day in the Life in the affections of Beatles fans, but it’s a better song than Free as a Bird or Real Love. And posthumously reworked as a Beatles track, it definitely packs a greater emotional punch. If you want to be moved, the lyrics provide ample space in which do so. It’s doubtful whether Lennon had his fellow Beatles in mind when he wrote the song – although who knows? – but with a new middle eight sung in tandem by Lennon and McCartney, it very much becomes a song about the Beatles, expressing a yearning for their bond: “Now and then I miss you / Now and then I want you to be there for me.” There’s something similarly moving about the sound of a very Harrison-esque slide guitar solo being played by McCartney, who apparently balked at Harrison’s slide guitar additions to the mid-90s sessions as too reminiscent of his 1971 solo hit My Sweet Lord. That was precisely the kind of older brother-ish judgement that always rankled with Harrison: there’s something rather touching about McCartney paying tribute as if in shrugging concession that he might have been wrong, although Harrison’s actual presence seems to be restricted to acoustic rhythm guitar.

Advances in technology have solved the problems with Lennon’s vocals, which are nothing like the spectral presence that floated through Free As a Bird. The other potential vocal problem – at 80, McCartney’s voice has aged considerably since the remaining Beatles last reconvened – is solved by keeping him low in the mix: you feel his presence rather than notice it directly. The additions to a song that was obviously incomplete are seamless – again, unlike Free as a Bird, where McCartney’s new middle eight jarred slightly against Lennon’s original song – the arrangement is sumptuously tricked out with orchestration, but never stoops to deploying obviously Beatles-y signifiers. If you squint, you could just about imagine that it’s the Beatles playing together, which definitely wasn’t true of the mid-90s songs.

So Now and Then is a qualified success, although the question remains: what’s it for? It clearly doesn’t exist to make money, which none of the Beatles or their estates need – although the 7in single version retails for an eye-watering £18 – nor to burnish the Beatles’ existing catalogue, which hardly needs burnishing. Perhaps the real reason for its existence lies with McCartney. No Beatle tried harder to keep the band together or seemed more shattered by their split. And no Beatle has worked more tirelessly to affix a happy ending to their story, never failing to remind interviewers that the band were a tight studio unit to the end, regardless of what was happening outside of it, and that he and Lennon were friends again at the time of his death; re-releasing the Let It Be album without Phil Spector’s orchestrations (an addition that McCartney called the “breaking point” in the Beatles’ demise in the 1997 book Many Years from Now); green-lighting the Get Back documentary series, which showed their 1969 recording sessions in a happier light than the baleful Let It Be documentary; using the same technology behind Now and Then to duet with Lennon onstage at Glastonbury. The premature conclusion of the mid-90s sessions clearly niggled him: he has repeatedly mentioned finishing Now and Then in the intervening years. Now he has, an act of closure underlined by one of the lyrics he appended to Lennon’s: after the lines about missing you and wanting you to be there for me, he adds “always to return to me”.