I know Coke, the third part of that holy trinity, was around & popular by the early 20th century, but how popular were burgers & fries? Would someone have regularly eaten them? I remembered that I don't recall any WWII films showing people eating them, but was that due to wartime shortages?
So we know that French fries have been around since at least the late 1700s - and we're also pretty sure that they originated in Belgium. Thomas Jefferson had a White House dinner at which "potatoes served in the French manner" were served. The Belgian recipe made its way to France, and then over to America, which is how it got the name. Fries were served in the 1800s at American cafes, usually as a snack, but to answer your question (finally), French fries were pretty commonly found on menus with burgers as far back as the 1920s (usually listed as "French fried potatoes").
As for the hamburger, there are conflicting origin stories, but the Library of Congress recognizes Louis' Lunch in New Haven, CT, as the birthplace of the burger. It was a hit when it was served at the 1904 World's Fair in St. Louis. However, when Upton Sinclair published The Jungle two years later, the hamburger took a hit - after all, ground beef could have anything in it!
White Castle (founded in Kansas in 1921) with its clean interiors, smart-looking cooks, and (most important) on-site grinding, rehabilitated the burger. In fact, White Castle is the inspiration for the other fast food chains that followed. So from the 1920s forward, burgers were pretty popular - you'd find them in diners and roadside stands all over the country.
During the Depression, meat was expensive, and so burgers would have a lot of fillers; the onion-fried burger (where 50% of the patty is actually onion) of Oklahoma dates from this period. Others would use crackers, flour, or bread to stretch the meat. Meat rationing at home during World War II further decreased burger consumption (fun fact: during the war, people took to calling hamburgers "Liberty Steaks," another chain in the proud tradition of stupidly renaming things because we don't like the people it's associated with), but for the soldiers? They got the meat, so they got the burgers. It's commonly assumed that American soldiers introduced the hamburger to the rest of the world.
So, yes, the holy trinity of burgers, fries, and a Coke (or a shake/malt) was firmly in place by the 1920s.