Suppose the Athenians, in order to maximize the usefulness of their navy, land in Sicily. A long series of skirmishes arises with Syracuse, which for several decades is fairly inconclusive. The Athenians make tentative overtures of friendship towards the Roman Republic, as the Athenians took their democratic rule very seriously (in OTL, they would have installed democratic regimes in many of their conquered foes or subjugated allies; it seems reasonable that they would ally with a somewhat democratic city for this reason). When the Persians invade from the north, beginning to shatter the Etrurian states, the Athenians make an offer: they will evacuate the Romans to Corsica, as the Athenians themselves fled to Sicily. After a stunning defeat at the hands of the Persians, the Romans accept.
When, a decade or so later, the Helots revolt in the Peloponnesus, followed swiftly by the Samians, Lesbians, and then the rest of the Greeks, the Athenians and Romans cooperate to retake the Italian mainland. Over the course of the next half-century, the combination of Roman and Athenian power manages to overcome Syracuse and the rest of the Greek cities of southern Italy; during this time the two nations have been growing closer together in culture and governance, though there's still considerable distinction. It is not yet time for Athens and Rome to be one and the same nation.
EDIT: The question will arise "What was Carthage doing?" I'd expect the answer to be that they were making further inroads in Iberia, and possibly spreading eastwards towards Egypt, as it's been made clear that the Persians can't hold onto it any longer.