Question: If Nisan is the first month of the year, why does the New Year begin in Tishrei?
Answer: The year begins on different dates for different purposes: in Nisan, for dating a king’s reign; in Elul, for giving tithes from cattle; in Tishrei, for numbering the years and for giving tithes from crops; in Shevat, for giving tithes from the fruit of trees (Mishnah Rosh Ha-Shanah 1:1). We number the months of the year from Nisan, but the number of the year itself changes in Tishrei. The fact that Nisan is the first month is explicit in the written Torah (Ex. 12:2), and the fact that the years are counted from Tishrei is in the oral Torah (an undisputed statement in the Mishnah).
Yours, Rabbi Azriel Schreiber