What is the ruling on picking up a valueless and broken watch from around the Kaaba and using it to know prayer times during the performance of Hajj rituals, then disposing of it after completing the rituals, and what is the expiation for this act, whether the picker was in a state of ihram or not?

The Sharia has emphasized the ruling on picking up lost money in Mecca. It is not permissible to take a lost item there except for someone who intends to announce it and continuously search for its owner, or to hand it over to the responsible authorities. This is to protect the wealth of pilgrims and 'Umrah performers, and because people disperse from Mecca to various places, making it difficult for the owner of a lost item to claim it.

Throwing the watch in the trash was another mistake, as it is a waste of money and does not absolve one from the sin of taking it.

It is not a condition for money to be of high value for one to search for its owner; rather, it is sufficient for it to have a value that its owner cares about and would inquire about.

Whoever picked up the watch must give its value in charity on behalf of its owner. The reward for this will be for the owner, and this will expiate your sin.