The story isn't true. It most likely stems from an inaccurate account by the English writer A. N. Wilson. It is also possible that Wilson's version has been 'blended' with a piece of fiction by a Dutch writer, Dick Raaymakers.
There is also at least one other false 'version' of the 'Mussolini, Laurel & Hardy & bowler hat' story.
The source of this apocryphal story is most likely the English writer A. N. Wilson's biography of Hillaire Belloc. Alan Brien's Diary in the London Review of Books cites it as
a precise example of stretching the evidence, pushing
the novelist’s licence to caricature beyond its legitimate limits, in
On Mussolini's dress style, A. N. Wilson wrote:
‘Mussolini had in fact modelled
his style of dress on that of his favourite film stars, Laurel and
Hardy, whose sartorial distinctiveness he regarded as the embodiment
of trans-Atlantic chic. He only abandoned it when he was informed that
Laurel and Hardy were generally appreciated, not as models of
sophistication, but as clowns.’
Brien asserts that
Wilson, by a Lytton Stracheyish sleight-of-hand, has turned the story
upside down for an easy laugh.
For evidence, he cites Denis Mack Smith’s Mussolini in which the author writes that Mussolini:
‘discarded his bowler hat when he noticed that in American comedy
films (for which he had a passion and installed a projector) they were
no longer worn except by his favourite stars, Laurel and Hardy.’
J. Kenneth Brody, in The avoidable war: Lord Cecil and the policy of principle, 1932-1935, also notes that Mussolini
gave up bowler hats when he noticed that in American movies, which he
adored, only Laurel and Hardy seemed to wear them.
Also, it is easy to prove that Mussolini did not model his style of dress on Laurel and Hardy: Mussolini was already using a bowler hat before 1914 when he was editor of Avanti!. This is at least seven years before Laurel and Hardy even made their first film together, and at least 13 years before a bowler hat was worn in one of the comic duo's films (Laurel wears one in the 1927 short Duck Soup, but Hardy has a top hat).
Mussolini and bowler hat in November 1922. Source: History for Sale
The first appearance of the bowler hat in a Laurel and Hardy film (Duck Soup, 1927). Source: The First Pairing of Laurel & Hardy: On the OTHER “Duck Soup”
Mussolini was somewhat notorious for his dress sense (or rather, lack of): for example, he often wore spats (which, again, were not copied from Laurel and Hardy) "long after the fashion had changed".
A. N. Wilson’s apophrycal story may have got mixed up with Dutch composer and essayist Dick Raaymakers’ fictitious 1984 story The Fall of Benito Mussolini: A Forgotten Hollywood Project of 1930, which was turned into a 1995 theatre production. Raaymakers’ story concerns an obscure director who tries to interest Hal Roach studios (makers of the Laurel & Hardy films) in a film on the fall of Mussolini using clips from the comic duo’s 1930 film Night Owls. This fictitious propaganda film contains the following:
In the Park scene at the beginning of Night Owls, it wouldn’t be
Laurel and Hardy who would awaken on the bench but Stan Laurel and
Mussolini, played by Oliver Hardy
The story was probably partly inspired by an attempt in 1937 by Mussolini’s son Vittorio to persuade Hal Roach to participate in a film project. The project never got off the ground as Vittorio’s visit quickly ran afoul of the Hollywood Anti-Nazi League, forcing Hal Roach to drop the idea.
Peter Neville, in Mussolini, claims that Mussolini
continued to wear bowler hats until the admired Laurel and Hardy
ceased to do so in the silent American films he enjoyed watching.
This is patently false; Laurel and Hardy wore bowler hats in their last film, Atoll K (aka Utopia, 1951), six years after Mussolini's death (and they stopped making silent films in the late 1920s).