The song - which was Cilla's first ever UK No.1 single waaaaaay back in 1964 - featured in the second episode of the hit drama about the singer and TV star's rise to fame.
More than six million viewers saw Sheridan Smith playing Cilla as she recorded the song under the watchful eye of The Beatles manager Brian Epstein and legendary producer George Martin.
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Sheridan Smith As Cilla
If the song does return to the chart, it would mark the first time the 71-year-old has been in the Top 40 since her 1974 release 'Baby We Can't Go Wrong', which reached No.36.
Shortly after the second episode of the three-part drama aired, 'Anyone Who Had A Heart' - which was originally recorded by Dionne Warwick - flew up the iTunes chart to No.52.
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Cilla Black
'Cilla' has been a huge hit for ITV. As well as huge critical acclaim for Sheridan Smith's portrayal of Cilla, it has been the channel's highest rating drama launch since 'Broadchurch' last year.
Cilla’s version of ‘Anyone Who Had A Heart’, one of her signature tunes, was named one of the biggest female hits of the 1960s by Radio 2. Dionne Warwick originally recorded the song, claiming Cilla had copied her rendition and adding: “I honestly believe that if I'd sneezed on my next record, then Cilla would have sneezed on hers too. There was no imagination in her recording.” (credit:PA/PA Archive)
Cilla Black(02 of07)
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Cilla may have given Simon Cowell a firm “no” when it came to appearing as an ‘X Factor’ judge, claiming she would find it too difficult to put down the less talented contestants, but she has some experience with TV talent shows, having judged the stars of ‘Soapstar Superstar’ in 2006. (credit:Julian Parker via Getty Images)
Cilla Black(03 of07)
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And speaking of soaps, Cilla finally achieved one of her showbiz dreams in 2013, when she made a guest appearance in the Rovers Return as part of a sketch for ‘The One and Only Cilla Black’, a retrospective on her 50 year career. (credit:ITV)
Cilla Black(04 of07)
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Other recent acting credits Ms Black has under her belt include a guest appearance in ‘Benidorm’, where she played an exaggerated version of herself. She was meant to be starring in a new sitcom ‘Led Astray’ with her pal Paul O’Grady, but ultimately the project was shelved after they complained they couldn’t handle the long hours. (credit:Ian Gavan via Getty Images)
Cilla Black(05 of07)
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Cilla became known in the late 1980s for her show ‘Surprise Surprise’, when her controversial decision to sing at the ending of each show divided the nation. Sadly, Sheridan Smith does not take on the challenge of this number during the ITV biopic about her life. (credit:Anwar Hussein/Anwar Hussein)
Cilla Black(06 of07)
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When asked why presenting had taken a back seat now, Cilla admitted during a ‘Loose Women’ interview that she’d rather see young and attractive people when she switched on TV, later telling The Lady: ‘I’d rather watch Holly Willoughby on TV than me.’ (credit:Steve Parsons/PA Archive)
Cilla Black(07 of07)
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She may be known for her light-hearted entertainment, but our Cilla’s not afraid to get political when she feels strongly about something. A staunch supporter of the Conservative Party, she once said Margaret Thatcher had “made Britain great again”, and appeared on stage at a Conservative Party Rally in 1992. (credit:Neil Mockford via Getty Images)