As usual you are correct: nutmeg in Bechamel is a long standing Italian tradition according to the Internet. Also, in Mastering the Art of French Cooking, Julia Child does not mention nutmeg in the basic Bechamel or Veloute sauces; it is not included until she give the recipe for Mornay sauce.. Where I picked up the nutmeg habit remains a mystery; no one in my family is even remotely connected to Italy. We are French, Irish, and German. I must have picked it up when preparing Shrimp Alfredo or something similar as a young married woman. I can't think of anything Italian I have ever made except Alfredo using a Bechamel. Besides spaghetti and meatballs or meat sauce and Alfredo, I really don't cook Italian.
When I thought back to laurenlulu's post earlier in this thread, the mystery may be solved. Adding nutmeg is something we in New Orleans do. Go figure. The French and Spanish influences in cooking were predominant when I was growing up. I did read that the French appropriated Bechamel from the Italians. We probably appropriated the nutmeg too.
Not that it is important, but one of my ancient college professors, from many, many years ago, did his thesis about how the same nationalities settled in both New York and New Orleans, which may explain the similar accents the two cities share. There were five primarily national groups. I don't know if I remember that ancient history, but besides French and Spanish were Irish, German, and (what?), maybe Italian.