Gaul not an Empire. It was not even a unified state. There were two Gauls: Translapine Gaul (France, Belgium Holland south of the River Rhine and Germany west of this river) and Cisalpine Gaul (in part of northern Italy.) The former was inhabited by some 54 independent major and minor tribes and the latter was inhabited by eight independent tribes.
Alexandria was not an empire either. It was a city. At the time in question it was the capital of a kingdom: the Ptolemaic kingdom of Egypt.
Carthage had an empire in the western Mediterranean: its homeland in Tunisia and western Libya, the islands in this part of the Mediterranean (Sicily, Sardinia, Corsica and the Balearics) and trading outposts in southern Spain. Following her first war against Rome she conquered southern Spain and the greater part of its east coast.
Before the wars against Carthage Rome controlled Italy through a system of alliances with some of the Italian peoples and through founding settlements in strategic areas of the peninsula. She started developing an empire outside Italy by winning the three Punic Wars against Carthage, taking over her territories after each war.
Hannibal did not march on Italy from Carthage. He marched from southern Spain, which his family had conquered. He marched along the eastern coast of Spain and inland from the southern coast of France. He then crossed the Alps to reach Italy. He took an army with some 58,000 men, the majority of which were cavalry, siege machines and war elephants.
Brittany
Gaul is the name the Romans gave to the area which is (mostly) located where is France today.
The Romans took over Tunisia and western Libya (Tripolitania), which had been Carthaginian territories, after they destroyed Carthage at the end of the Third Punic War in 146 BC.
The Romans called it Gallia - Gaul; to be precise, Gallia Transalpina, Gaul beyond the Alps.
Gaul
Carthage was a settlement for retired soldiers.
Gaul wasn't a powerful empire.
The struggle with Carthage required the development of a naval force to augment Rome's land force, and the 120 year struggle established Rome's dominance in the Western Mediterranean and beginning its involvement in the Eastern Mediterranean against Macedonia. The struggle to dominate Gaul extended Rome's commitments from southern Europe into the north, expanding its now standing military force to conquer and then garrison Gaul then Britain, and coincidentally launched Julius Caesar into dominance in Rome.
Romans,Carthage's Egyptians Gaul's Spain, Aztec's Persia, the Greek Cities India China, Mongolia, Japan. Want me to keep going?
There were Jewish communities throughout the Galilee (Israel); Rome, Bari, Otrento, Lucca (Italy); Cyrenaica (Libya); Alexandria (Egypt); Athens (Greece); Carthage, Antioch; Marseilles (Gaul); Sicily; Cyprus; Tyre, Sidon; Spain; Rhodes; Turkey, and more.
Edward Gibbon (The Decline and Fall of the Roman Empire) says that by the early part of the fourth century, Christianity was strongest in the eastern Empire and in the province of Africa (based on Carthage), as well as in Rome itself. He says that Christianity continued to struggle for converts in Gaul and Britain, and that outside Alexandria, the Egyptians showed little interest in Christianity.
Julius Caesar
There were three Punic Wars between Carthage and Rome, and Carthage lost all three of them. In the first war, they lost their territory in Sicily. In the second war, the one in which they came closest to winning, they lost their territory in Spain, Gaul, and Africa. In the third war, Carthage was destroyed after most of the Carthaginians had starved to death in a three-year siege. The survivors were sold into slavery, and the city was destroyed.
Caesar as a powerful general conquered France & Belgium, then called Gaul. Gaul was added to the Roman Empire. Julius Caesar also invaded Britain. Before his murder, he destroyed the army of his rival Pompey in Greece. Caesar while in total command of Rome, granted his allies in Spain & Gaul, Roman citizenship.
Initially it raised them from its citizens. However in 480 BCE it suffered a massacre in Sicily when the Greeks ambushed 10,000 of them crossing a swampy river. Thereafter they relied on mercenaries from North Africa, Spain ad Gaul.
Gaul