Gunpowder Empire, Tor, 2003, is the first novel in the Crosstime Traffic series. It follows the adventures of the Solters family, who spend every summer in the alternate universe called Agrippan Rome, more precisely in the city of Polisso, in the Dacia region, which is Romania in their reality. Agrippan Rome is a world where the Roman Empire never fell, thanks to Marcus Vipsanius Agrippa's survival against disease, which let him lead the Roman armies against the invading barbarians.
The Solters setup shop in Polisso, selling pocket watches ("hour-reckoners"), sharp razors, mirrors, and Swiss knives, all trinkets slightly above the technological know-how of the people. Things start going awry when Mrs. Solters falls ill. She and Mr. Solters leave for the home timeline, leaving their children Jeremy and Amanda behind. All would be well, except that soon after Jeremy and Amanda lose the communication link to their home timeline, and the neighbouring kingdom of Lietuva (Lithuania and possibly parts of Russia in the Solters' home reality) has once again declared war against Rome, and the siege of Polisso begins.
The book's plot has much in common with Household Gods, written by Turtledove in collaboration with Judith Tarr - though that book concerns time travel back to the actual Roman Empire. Both books depict the daily life of a provincial Roman city from the point of view of modern American(s) marooned there, and both deal prominently such issues as facing a society where slavery is taken for granted, and surviving a foreign invasion of the Imperial territory.
In "Alternate History: The How-to of What Might Have Been," Turtledove says on coming up with a story using the Roman-German conflict of Augustus' day as a break point: "It took me more than twenty years to come up with a story to follow on changing things there."