The modern Hebrew calendar differs from the ancient calendar observed by Moshe as leap months were added and names added Nissan, Elul, etc. If one were to devise a calendar based on the ancient calendar alone and using the months 1,2,3 etc. what would it look like and when would the holidays fall?
It would be identical with our calendar, which is based upon the lunar month and solar year, as detailed extensively in the Oral Tradition. The only difference would be the names, as you mentioned, where we see the Torah calls Nissan the first month, and then the 2nd, etc. but we see that there were also original Hebrew names, some of which were not lost, for example, Eitanim=Tishrei, Bul=Marcheshvan, Ziv= Iyar. Regarding the order, Nissan was always the first month counting since the exodus from Egypt, and so it continued regarding the holidays, and how kings count their reign, while Tishrei is counted first since creation regarding general years, shmita etc.