Yes, LSD has a history of psychiatric use
In terms of the number of doses administered in an official clinical setting, it might not fit the definition of widely, but it has been tried out for wide number applications.
From its first synthesis in 1938 to Bicycle Day in 1943 (when the developer Hofmann discovered its mind altering effects by accident) there was little usage. Since then, the manufacturer Sandoz explored possible clinical applications and eventually took it to market as Delysid:
Variants of this product in images:
The package insert leaflet from ~1950:
— Albert Hofmann, LSD, mein Sorgenkind, Stuttgart 1979. (partly on the web)
The 1964 Sandoz catalogue entry for Delysid lists as applications in English language:
Delysid is used in analytical psychotherapy to elicit release of repressed material and to provide mental relaxation, particularly in anxiety states and obsessional neurosis […]
Delysid intensifies the reactions of psychotic patients, and useful information can be obtained by its use in selected patients. In certain forms of psychosis, particularly schizophrenia, and in chronic alcoholism, high doses may be necessary to produce the typical response to Delysid. […]
Throughout the 1950s research into LSD and eventual clinical practice grew substantially, until, as a DEA-report puts it:
Sandoz Laboratories, the drug’s sole producer, began marketing LSD in 1947 under the trade name “Delysid” and it was introduced into the United States a year later. Sandoz marketed LSD as a psychiatric cure-all and “hailed it as a cure for everything from schizophrenia to criminal behavior, ‘sexual perversions,’ and alcoholism.” In fact, Sandoz, in its LSD-related literature, suggested that psychiatrists take the drug themselves in order to “gain an understanding of the subjective experiences of the schizophrenic.”
In psychiatry, the use of LSD by students was an accepted practice; it was viewed as a teaching tool in an attempt to understand schizophrenia. From the late 1940’s through the mid-1970’s, extensive research and testing were conducted on LSD. During a 15-year period beginning in 1950, research on LSD and other hallucinogens generated over 1,000 scientific papers, several dozen books, and 6 international conferences, and LSD was prescribed as treatment to over 40,000 patients.
— DEA: "LSD: The Drug" (web-archive capture from 27 Apr 1999 - 22 Sep 2021)
Wikipedia summarises this report:
Sandoz Laboratories introduced LSD as a psychiatric drug in 1947 and marketed LSD as a psychiatric panacea, hailing it "as a cure for everything from schizophrenia to criminal behavior, 'sexual perversions,' and alcoholism."
The movie in the question is an official government production found at the Imperial War Museum:
Record of the Moneybags Trial (abbreviated version of Small Change) when volunteer subjects from 41 Royal Marine Commando engaged in a field exercise after imbibing the hallucinogen LSD (lysergic acid diethylamide) at Porton, 27 November - 4 December 1964.
— "A Trial of an Incapacitating Drug [Main Title]", Imperial War Museum, Catalogue number, MGH 4464