What is the ruling on one who relied on an athan application for fasting, then it became clear that the time for the athan of Fajr in it was one minute more than the original time, so he drank one minute before the athan?

If it is merely a doubt about eating or drinking after the appearance of dawn, then the fast is valid, because the original state is the continuation of night. However, if you are certain that you ate or drank after the appearance of dawn, then you are required to make up that day according to the majority of scholars. You are not required to make up that day according to the chosen opinion of Sheikh al-Islam Ibn Taymiyyah, but making up that day is more cautious.