Where I grew up, in Poughkeepsie, NY, in the era stretching from about 1939 to 1950, there was lots of fast food - altough we never thought of it in that exact term.
Every saloon had a free lunch on the bar - wursts, salads, boiled eggs, pickled eggs, pickles, cold meats, roll mops, bread, pretzels, clams, oysters, and much more. Not every saloon had all of the items listed above - the choice in any given place was usually quite limited by ethnicity, season, or just the whim of the saloonkeeper. This was really fast food - men (very very few women) would rush in, chug down a shot and a beer , grab some free lunch and rush out again. When I was about 12 years old my father started to occasionally take me to a bar and I'd load up with whatever was available while he gabbed with his friends, placed a bet on the horses, and had a beer. I can't say children were especially welcomed but they were tolerated if they behaved.
We also had several diners and lunchrooms where you could get a quick meal (take-out was available in some places, but not commonly used). The quickest meals were things like a bowl of stew or soup, a cold sandwich, a hot open-face sandwich or a slice of cake or pie. Unless we had to wait in line they could always get us in and out in 10 minutes or less- how fast do you want?
And, of course, there were food carts and trucks serving hot dogs and hamburgers and cold sodas and such. It seemed like any event that could draw fifty people would be sufficient to bring out the carts and trucks.
The only specifically fast food place I remember was the White Castle on Main Street. As I remember they only served those little square hamburgers and french fries - nothing else on the menu - and they were very fast.
Pizza (apizza in those days) was not considered fast food. Alloys ( dating from 1928 and still in business and still very good) was the only pizza joint I was aware of until about 1950 when the whole pizza craze really started to boom. At Alloys we ordered our pizza (a square pie), sat at the bar or a table and waited about 15 minutes while the pie was freshly prepared. ( I think we could usually get a slice of pizza served immediately at room temperature - I guess that's about as fast as fast food gets.)
Every grocery store (we did have two or three proto-supermarkets) had a display of little pies and such - wrapped in wax paper. They were a popular source of a quick snack.