An easy way is to have the Phoenicians penetrate deeper into Italy instead of the Greeks. Rome adopts the cult of Baal, Ashera, Astarte, Tanit and Melkart and the Phoenician abjad instead of the Greek alphabet. Just have Tyrians found Carthage in Sicily or southern Italy instead of Tunisia.
As for the consequences, the Greeks may have gone the way of the Phoenicians, with little of their literature surviving (indeed, no Phoenician literature survived except for a translation of Hanno's geographical work, and the closest to native Phoenician literature that isn't the Bible are the Ugarit tablets preserving the myths of Baal and Anat) and even other famous elements like sculpture and architecture as the Romans would not have preserved those as much as it did originally.
This means the continent of Europe would not be called "Europe" since Phoenicians had others names for it, and "European" intellectuals of later eras would not be looking towards Greece but towards Phoenicia and Italian Carthage. In terms of religion, though, something like Christianity could still become the religion of this alternative Phoenicianised Roman Empire, and something like Islam could still conquer over half of this empire like in the OTL.
If this Phoenician ATL "Europe" also still goes the way of colonialism, then Phoenician culture and civilisation would predominate just like Greco-Roman civilisation does right now.