In this case, the easiest thing to do is actually just to download the previous save and compare it to the current one. If you let your mouse pointer hover over a tech in F6, it tells you how many turns it would take to research that tech. Any techs that take fewer turns now than they would have taken last turn, Team Doughnut must know.
Alternatively, you can look up the base tech price in the editor. Multiply the tech price by the map modifier (30 for a small, emperor level map), and you get how many beakers it takes to research that tech if nobody knows it. Multiply the number of beakers per turn you are producing (add them up in F1) by the number of turns it will take you to research it, and you get the number of beakers you actually need. From this, you can calculate how many other civs know that tech.
Alternatively, you can look up the base tech price in the editor. Multiply the tech price by the map modifier (30 for a small, emperor level map), and you get how many beakers it takes to research that tech if nobody knows it. Multiply the number of beakers per turn you are producing (add them up in F1) by the number of turns it will take you to research it, and you get the number of beakers you actually need. From this, you can calculate how many other civs know that tech.