Let's say that we would still exist, because most of the ancestors of present-day Hungarians are not Asians, but Slavs, Germanics, and other European peoples, so you yourself would continue to exist even in this alternate world.
The conquering Magyars never reach the Carpathian Basin, even the Huns avoid it.
Under the rule of Emperor Traian, Pannonia Province flourishes to a greater degree, than in our world, and in later periods, fewer Barbarian invaders get here.
The Roman Empire still falls, but Pannonia remains as a sovereign country, as the other Neo-Latin countries did.
By the 21st century Pannonia is the Neo-Latin country that has best preserved the culture of Ancient Rome. Although Christianity was forced upon the people in the Middle Ages, by our present time many returned to Roman Polytheism. They believe in Jupiter, Mars, Minerva, Bacchus and other gods. Saturnalia is a more popular holiday than Christmas.
In warm weather, many people walk around in togas, Homosexuality is acceptable, and there are many great feasts and parties with wine.
The walls of houses are decorated with sex scenes, street lamp poles are topped with a Phallus.
Through the times, Latin was replaced by the Pannon-Romanian language which sounds like a mixture of Romanian and Portuguese with a heavy Slavic influence, we speak this. In more educated circles, in universities, and courts, being fluent in Classical Latin is a tradition that still holds.
The capital of Pannonia is Kestei ("Castellum" in Classical Latin, "Keszthely" in our world), there is even a local Colosseum and Circus Maximus here.
Our flag consists of a blue and green stripe, with a stylized Pelso (Lake Balaton) in the middle.
Our anthem and language sounds like this:
How would you like if we lived like this?