A lot of names have been tossed around over the years regarding who the headless man could be. When the government initially commissioned Lord Denning to investigate the issue, per Wikipedia, he made a list of five potential suspects based on handwriting on other Polaroid photos included in the set. These men were actor Douglas Fairbanks, Jr., former Minister of Defense Duncan Sandys, businessman John Cohane, Savot Hotel press officer Peter Combe, and German diplomat Sigismund von Braun.
But a later documentary titled Secret History: The Duchess and the Headless Man claimed to have identified not one, but two men who are behind the infamous role. Casting aside a lot of the notions of the original investigation, the movie claimed that Douglas and Duncan were named as the most likely suspects, per The Guardian.
This conclusion was drawn using evidence from unpublished parts of a report from Lord Denning, who was first assigned to investigate the case as well as testimony from Margaret’s friend Paul Vaughan. Paul reportedly explained that Margaret told him at the time that "the only Polaroid camera in the country at this time had been lent to the Ministry of Defense."
However, the real identity of the headless man was called into question yet again in 2013 when Lady Colin Campbell, the Duke's daughter-in-law, identified the headless man as William "Bill" Lyons, sales director of Pan American World Airways, per The New York Post.
Considering the fact that Margaret took the true identity of the headless man with her to the grave, it's unlikely that royal fans will ever get a confirmed answer to that ever-burning question.