The two traditions used different power sources, a water clock feeding an escapement-like linkwork versus a purely weight-and-spring drive train, so the answer turns on which feature defines "mechanical clock," not on a missing fact.
What would resolve this A settled scholarly definition of "mechanical clock" that either includes or excludes an escapement-regulated but water-powered drive train would resolve which tradition holds priority.
OpenHistory of technologyWikipedia contributors, History of Timekeeping Devices (Wikipedia)