There are some answers on individual aspects of this question, and it's important to remember that the decline of paganism didn't happen all at once around the world.

Why did Christianity, and not another contemporaneous cult, spread so rapidly within the Roman Empire? and How significant was Constantine's conversion to Christianity? by /u/talondearg

Assuming an atheistic perspective, what factors gave rise to the widespread adoption of Christianity in ancient Rome prior to it being made the official religion of the empire? by /u/Zosim

How late did Greco-Roman polytheism / paganism last as a religion after the Roman Empire became officially Christian, and where were the last areas it was practiced?

How did the conversion from Norse pagan religion to Christianity work? Do we have any personal accounts? by /u/Steelcan909

How prevalent was Scandinavian paganism in Viking-Age Britain and Ireland? by /u/textandtrowel

What was the religion of Egypt like after Cleopatra and before Christianity? by /u/cleopatra_philopater

When the Anglo-Saxons Kings were first converting to Christianity were most of the peasants still pagan for a while after, or had they already mostly converted? by /u/wizzyhats

What was the culture and religion of the Arabian Peninsula like before Islam? by /u/Daeres