I've written about this before so will repost that with some minor tweaks.

During World War II, Hollywood directors such as John Huston and John Ford volunteered for service with the military (Army, and Navy, respectively), where their talents were put to the best use, namely making movies. They, and others, were tasked with creating films for domestic consumption to bring the war home to American audiences, in other words, propaganda. Ford, for instance, shot the film "Battle of Midway", during the fight itself, although in part it was sheer luck he was present, the Navy having already sent him there to shoot more tranquil footage. He recalled the irony afterwards that "I think at the time there was some report of some action impending but [...] I didn’t think it was going to touch us. So I [...] spent about 12 hours a day in work, had a good time up there.”

By far most famously though was D-Day. The Allies intended to thoroughly document their triumph there, and several hundred ships were equipped with movie cameras, as were some 50 landing craft, all constantly rolling and not needing human touch. In addition, Ford , along with George Stevens, was given film crews to be sent on in with the troops, many with color Kodachrome and part of a documentary force that numbered in the hundreds when you include still photographers (Ford was attached to the Navy and OSS, Stevens to the Army, and coordinated very little). Stevens run shooting at Juno Beach with the British (Yes, Canadians landed on Juno. Stevens was working off of a Royal Navy ship), while Ford was in charge for Omaha, where the most intense fighting was. He would actually refuse to talk about it for many years after, but did eventually offer his recollections, although they come with a grain of salt:

Once I was on the beach I ran forward and started placing some of my men behind things so they’d have a chance to expose their film. I know it doesn’t sound blazingly dramatic. [....] To tell the truth I was too busy doing what I had to do for a cohesive picture of what I did to register in my mind. We stayed on the job and worked that day and for several other days and nights too.

Ironically, for all their efforts, almost none of the footage would see the light of day, for several unfortunate causes. In the first, much of the footage was destroyed, packed into a duffle-bag that was accidentally dropped into the water! As for what little remained, after being processed in London a few days later, a ~100 minute highlight reel was sent to several leaders such as FDR and Churchill, but kept tight under wraps, at least according to Ford "[a]pparently the government was afraid to show so many casualties on screen." The camera crews hadn't held back, and it was simply too real for the people back home, it would seem. It was not entirely lost though. Those familiar with the story from Stephen Ambrose's book on D-Day will likely only know the conclusion at that point, when no one knew where the footage was still, but since that time, some of it was, in fact, rediscovered in US government storage, although just how much remains unclear, and almost none of that has ever been released

As detailed in this 2000 article from Time:

Yet somehow Ford's footage was lost until 1998, when Melvyn R. Paisley, a World War II aviator and Reagan-era Assistant Secretary of the Navy, found a few canisters of the missing film deep within the National Archives. Spielberg, whose father had also served in the U.S. Army Air Corps and who would win the Best Director Oscar for his own D-day movie, Saving Private Ryan, was intrigued when he read about Paisley's find in the New Yorker.

So intrigued, in fact, that he recruited documentary maker and TIME film critic Richard Schickel and immediately started hunting for more raw footage from World War II. With Paisley's help, they amassed 600 hours' worth and began editing the remarkable trove. The result is Shooting War, a spectacularly conceived and haunting 90-min. documentary that premieres June 5 [2000], in conjunction with the opening of the National D-Day Museum, at a conference sponsored by the Eisenhower Center at the University of New Orleans.

The Shooting War, at least, is on YouTube, but it is about war photographers during WWII, not D-Day, so the actual D-Day footage is only a small part. This is the timestamp for when D-Day starts up. What parts are actually Ford's team's footage seems to be unclear, as much of it was legitimately lost, Paisley only recovering part of it. This I think is when it starts, as as you can see, it is quite brief, shot by a cameraman who was wounded and evacuated early on, presumably keeping his footage out of the cursed duffel.

Other D-Day footage of course exists. The aforementioned George Stevens, similarly tasked to run shooting at Juno Beach, also has surviving footage, some of which is excerpted here for instance. But the Omaha footage is almost entirely lost to the ages.

More broadly focused on the Hollywood men who contributed by shooting the war, "Five Came Back: A Story of Hollywood and the Second World War" by Mark Harris is a book on the topic, focusing on Ford and Huston, as well as Frank Capra, George Stevens, and William Wyler, who were also involved in the war effort.