Although the film was released in 1952, it wasn't shown in Los Angeles until 1972 & therefore wasn't eligible for an Oscar until then. The award was presented to Chaplin and his collaborators Raymond Rasch & Larry Russell, except the latter had nothing to do with the film. This was evidently a flub on the part of the Academy. By 1973, both Larry Russell and Raymond Rasch were deceased (Rasch's son and Russell's daughter accept the awards on their behalf at the ceremony). When the Academy asked Chaplin who arranged the music, he answered Raymond Rasch. When Rasch's widow was asked who else arranged the score with her husband, she said "someone named Russell." The Academy immediately thought of Larry Russell, who was also a music arranger. When the Academy asked Russell's widow if he had worked on the film, she only said "he must have." But he didn't. According to David Robinson's Footlights & The World of Chaplin's Limelight (2014) a letter exists in the Chaplin Archives that shows that, before work on the film began, Russell had offered his services as conductor, but they were declined and at no time was he ever employed by the studio.* It appears that the award should have been given to composer/arranger Russell Garcia. In an interview in 2008, Garcia was asked why he never made an effort to correct the mistake himself: "I don't want to make trouble for anyone or spoil anyone’s fond thoughts or memories...I've won plenty of awards. I just forgot about it." Read more of his interview here. Garcia passed away in 2011.
Strangely enough, Robinson's book also notes that Garcia's name appears nowhere in the daily records of the Chaplin Studio. "If he worked on the music, it can only have been a purely private arrangement between himself and Rasch." While this might be true, a photo does exist of Garcia & Chaplin at a recording session for Limelight.
Below is a photo from my collection, taken at the same time as the photo above. I thought the man next to Chaplin (in the white shirt) might be Garcia as well. I might be wrong but the hair and shirt are similar. The man at far left is Raymond Rasch.
*According to Robinson, following Russell's nomination in 1973, his widow "asked for a one-third share in performance royalties in the Limelight music--a claim which she quickly retracted, saying 'that she had made her claim due to a misunderstanding."