Crosstime Traffic, book 1: Gunpowder Empire
Harry Turtledove2003

Jeremy Solter is a teenager growing up in the late 21st century. During the school year, his family lives in Southern California-but during the summer the whole family lives and works in the city of Polisso, on the frontier of the Roman Empire. Not the Roman Empire that fell centuries ago, but a Roman Empire that never fell.

For we now have the technology to move between timelines, and to exploit the untapped resources of those timelines that are hospitable to human life. So we send traders and businesspeople-but as whole-family groups, in order to keep the secret of Crosstime Traffic to ourselves.

But when Jeremy ducks back home for emergency medical treatment, the gateways stop working. So do all the communication links. Jeremy and his sister are on their own, Polisso is suddenly under siege, and there's only so much you can do when cannonballs are crashing through your roof...