Update.
I took my watch and the letter I received into a Cartier boutique. I was told it would probably be several days before my watched had its check up and made plans to pick it up in a couple of weeks.
About three hours later I received a text letting me know that my watch was ready for pickup. I was so surprised that I called the boutique to verify that the text was not a mistake, and was told that my watch was indeed ready for pickup!
When I picked up my watch, I expressed that I was pleasantly surprised by how fast the service was. The store associate that was helping me told me it had been a slow day for repairs, and since my watch is a quartz the check up could be completed quickly.
I am very pleased with the service that I received from Cartier. The complementary watch check up was definitely worth the time spent dropping off and picking up the watch. Also, it is a great excuse for going to Cartier and looking at all their gorgeous jewelry!