A watch would have been handy for someone who rode the bus in the days before cell phones. A Benrus wristwatch and Elgin pocket watch that belonged to Oswald was apparently recovered from the Paine home.
https://texashistory.unt.edu/ark:/67531/metapth340212/m1/69/
From "Marina and Lee": "Marina noticed, out of a corner of her mind, that Alik kept the scar [on his left wrist] hidden. He wore a watch on his left wrist and once, when it was broken, it was months before he would take it to be fixed, Later, when he broke it again, and again had to take it to be fixed, he covered his wrist with an identification bracelet even though he hated jewelry and knew that Marina, too, hated jewelry of any kind on a man."
In the arrest photos he's wearing a bracelet on his left wrist. I wouldn't think he'd wear both a watch and a bracelet on it?
List of items found on him when arrested:
https://texashistory.unt.edu/ark:/67531/metapth340292/m1/1/?q=Lee%20Oswald%20moneyNo mention of a watch.