What is incumbent upon me after losing my father's watch, which I borrowed from him?

In this case, the watch is considered a borrowed item ('ariya). If it was lost due to your negligence, then you are liable for it according to the consensus of scholars. If it was lost without negligence, the preponderant opinion is also that you are liable. Its compensation would be by providing a similar item if one exists, or its value on the day it was lost if no similar item exists. Since the matter concerns your father, it is best to inform him of what happened and ask for his forgiveness.