According to this unidentified note Films and Filming (1989) [snippet view], the original source of the quotation is the Fench director Robert Bresson (who presumably said it in French):
...It's a cliché. Bresson said it first many years ago; you do make a film three times and the first time you make it on the page and that means the same stress that you have when you are shooting and when you are cutting. It's not a gentlemanly process and it's full of insecurities, worries and doubts for both sides.
This claimed origin also appears in Michael Ondaatje & Walter Murch, The Conversations: Walter Murch and the Art of Editing Film (2002) [combined snippets]:
Murch likes to quote Robert Bresson to the effect that a film is born three times—in the writing of the script, in the shooting, and in the editing. With The English Patient there were, in fact, four births, because there was also a book as the source.
And again in Sam Gregory, Gillian Caldwell & Ronit Avni, Video for Change: A How-To Guide on Using Video in Advocacy and Activism (2005):
A film is born three times. First in the writing of the script , once again in the shooting , and finally in the editing. ( French filmmaker Robert Bresson)
Earlier in the quotation's existence, however, the attribution seems less clear. For example, from Ray Hiebert & Carlton Spitzer, eds., The Voice of Government (1968) [combined snippets]:
The ability, courage, and judgment to make changes are assets to the film maker. A technical advisor who had just been through his first experience in motion picture production told me: “I've discovered that a film is born three times," he sighed. "Once when it is written. Again when it is directed. And once more when it is edited."
There is no match for "Bresson" in this text.
From an interview with Russian director Sergei Bondarchuk in Culture and Life, issues 3–12 (1971), [snippet view]:
I would put it this way: a film is born three times. The first time on the scriptwriter's desk, the second during shooting, and the third time in the cutting room. René Clair, the well-known French director, put it differently. He said: "The film is ready, the shooting is all that remains to be done."
Other, somewhat later sources have attributed the quotation to Francis Ford Coppola (2004), Jean-Luc Godard (2007), and John Sayles (2008)
The expression in its earliest form (in English) seems to have begun with the words "a film is born three times." It is recorded in a 1968 essay (which attributes it to "a technical advisor" to a filmmaker) and in a 1971 (where the speaker is Russian director Sergei Bondarchuk). The person most frequently cited as having originally made the obsrvation is French director Robert Bresson—but the earliest specific attribution to Bresson that I've found in print is from 1989. Most of the subsequent Bresson attributions cite Walter Murch and/or Michael Ondaatje as the person(s) who identified Bresson as the original source.
Bresson certainly might be the original source—and the fact that he might have made the observation in French in an interview or conversation not translated in its entirety into English would explain why nailing down the original quotation in an English-language source is so difficult. But it is also possible that Bresson isn't the originator of the observation. The only fact that I can point to with some confidence is that someone had said it—and someone else had recorded it in print—by no later than 1968.