So after seeing the Battlefield 1 reveal, it seems that many of the players were using sub-machine guns, LMGs and that sort. Excluding guns like the Vickers, how common were automatic weapons during the war? When did they become more common, and is Battlefield's abundance of them accurate?
They became very, very common. Just off the top of my head:
the French RSC M1917 was one of the first semi-auto rifles to be fielded in large quantities.
Browning BAR debuted in 1918, didn't see much use until the last few months of the war. Sort of bridges the gap between light automatic rifles and light machine guns, but much more a rifle than a machine gun. Crappy sights and being heavy were hinderances.
The American-designed, British-fielded Lewis Gun got used in large numbers, as an easy way for troops to boost their firepower in the very front lines. Light enough to be moved around by one guy, able to be fired from the shoulder if necessary. Oftentimes the gunner would have an assistant to carry extra ammo.
The Madsen light machine gun actually debuted in 1902, and is a pretty darn good gun. Used by various players in the war, in various calibers, and is still actually in use with the Brazilian Military Police
The MP 18 is considered by most to be the first true military sub-gun put into service. It was designed as something for trench raiders to carry so they could lay down a lot of automatic fire, but is much lighter than say, the MG 08/15, one of Germany's few portable automatic weapons of the time. Chambered in a pistol caliber, able to use snail magazines or stick mags, it let a soldier lay down a lot of fire very quickly. A whole host of small variations and design evolutions and competing models sprung up around it, but this is the one.
The MG 08/15 was just a chopped down, more portable version of the mostly-stationary MG 08. It still weighed like forty pounds, which is immensely heavy to run around a bunch of shell craters and trenches. Standard practice was to have an assistant or two, to help reload and carry extra ammo.
While it didn't get used very much at all, the Pedersen Device was a drop-in modification to the American 1903 Springfield rifle, giving it the ability to fire a 7.62mm pistol cartridge semi-auto.
There are a lot of other models that saw service in the Great War, but those are some of the bigger players off the top of my head. Long story short is that at the beginning of the war, automatic weapons were present and in-issue, but not that many compared to the number of troops fielded; most everyone on the front lines was serving a crew weapon or armed with a bolt-action rifle. The doctrine of 1914 (on all sides) had envisioned combat being something that happened at 500+ yards, with individual troops taking aimed, deliberate fire and employing good marksmanship to drill their target right dead center. Quickly, they found that wasn't the case; the proliferation of trench warfare proved that most combat was happening at less than 300 meters. This spurred development, adoption, and widespread use of light automatic weapons.
I haven't seen the second trailer yet, so I can't say how "accurate" it is; Ian over at Forgotten Weapons has a nice 10-minute breakdown of his observations. However, realize that a whole metric crapton of individual automatic weapons being fielded at the squad level. Just an example: in 1915 each British battalion had only four Lewis guns. By 1917, this number had jumped to 46, which is about one per section.