Well, it's not as if Shakespeare himself didn't update the settings of his own plays. "The clock hath stricken three?" They didn't have clocks that struck the hour in Ancient Rome! His portrayals of historical times and places were generally more like the world he and his audience lived in.
I believe the Romans had clepsydra (water clocks) that could strike the hour.
The costumes, props. etc used in Shakespeare's day are known to have been contemporary so, yes, they weren't obsessed about historical accuracy. The play's the thing.