Getting back to the original question...
Alexander Rose's book "American Rifle, A Biography" has quite a bit of fascinating insight into this question. There are many opportunities for divergence and the M-16's adoption is actually pretty unlikely (except in retrospect.) When it was adopted we had only a few combat advisors in Vietnam, far less than we had troops deployed in West Germany, South Korea, etc. with much longer shooting ranges and an expectation of moving by truck or armored personnel carrier rather than walking dozens of miles through heavy woods or jungle every day. 7.62 NATO was the standard round for our allies and we expected to be fighting side by side with them against the Soviets, Stalin's death and Gene Stoner's development of the AR-15 overlap and/or Chinese Communists (10 years after Korea.) Vietnam's expectations at the time of the rifle adoption are far in the future (and the M-1 and M-2 carbines (M-2 having full-automatic capacity and a 30 round clip often attached to a second for 60 rounds capacity were extremely popular in Vietnam and many of it's original use as the gun for non-frontline combat infantrymen.) Retention of the M-14, a triumph of Army Ordnance and Springfield Armory (who'd usually won these contests against externally designed, lighter, high volume carbines since the Volcanic, Spencer, Henry, etc.) would be the most likely scenario with the Thompson submachine gun, the "grease gun" submachine gun, and the M-1/M-2 carbine family filling the short/handy/automatic fire requirement...already owned in quantity, deployed all over the world, already in the ammunition and parts supply system as well as training.
The original application of the M-16's/AR-15 is General Curtis LeMay of the Air Force ordering these for his SAC-base/installation guards, not for Army jungle advisors. If Colt's barrel rifling twist rate not gotten misunderstood for the original test batch of the M-16's sent to Vietnam for combat testing, a rate of 1 in 16" instead of 1 in 7-9" that sent the bullets tumbling end over end for horrific wound effectiveness at close range, the rifle probably wouldn't have passed field testing. Soldiers deployed in Iraq were reporting not only 5-8 rounds often being required to kill an opponent but not penetrating heavy layers of clothing or car windshields, let alone mud brick walls, wooden doors, conventional civilian vehicles' doors, etc. has brought back M-14's into greater deployment as well as combat shotguns and interest in the 6.8mm SPC and other heftier cartridges for the platform.
Would have been a less desirable fit for Vietnam, just as no used M-1 Garands or 1903 Springfields in the Pacific Island campaigns against the Japanese, right? Just as the early days of Vietnam saw the same mix of 18" barrel/extended mag pump 12 gauge shotguns, M-1/M-2 carbines, Thompsons, etc. as the Pacific War saw so heavily, it would have just continued. The 17 pound Browning Automatic Rifle and 1919 A6 .30 Caliber Browning light machine gun on a bipod were extremely popular with jungle fighting in World War II so weight, length, and ammunition weight get a tradeoff many soldiers are willing to make. Probably wouldn't have changed Vietnam significantly although the soldiers would have had more confidence in their weapons which matters a lot in combat performance and a significant advantage in penetrating all of the forms of cover, like tree trunks, in those jungles while outranging the AK where there were clear areas. Being able to shoot the charging enemy long before they can shoot you is really handy, especially when overrunning a position is the standard approach.
While it's an easy point of divergence, the consequences aren't really coming to mind for the era, other than some American soldiers who died would have lived, come home and made their own impacts in the timestream so one could argue a key leader, inventor, etc. among them as there have certainly been tens of thousands of very high potential people lost in every war to chance.