It seems to have happened pretty early on in Church history, within the first 200-300 years or so, particularly after Constantine legalized Christianity and the Church began to set down rules about iconography.

So, the earliest visual depictions of Jesus that are 1) in color, and 2) show non-Arabic features, date from about 300 C.E.

Christ as Sol, c. 300 C.E.

Christ With a Beard, catacomb of Commodilla, Late 4th century C.E.. Wikimedia also claims this is the earliest known depiction of Christ with a beard, but take that with a grain of salt.

Christ Pantocrator, 6th century C.E.

That's not to say that darker skinned depictions of Jesus weren't around after Constantine... Take the mosaic in the apse of the Basilica dei Santi Cosma e Damiano, in Rome, c. 530 C.E. The Jesus depicted is decidedly dark skinned, especially when compared to the St. Peter, to the right (Jesus' left) and the pope who commissioned the piece, Felice IV. Also note the beautifully rendered and clearly Arabic features of Saint Cosmas (to Felice's left) and Saint Damian (St. Peter has his arm around his shoulder). What is perhaps just as interesting as the fact that the artist rendered Jesus with dark skin is the fact that he also rendered St. Peter with light skin and looking VERY Roman Empire. The historical Peter would have had similar features to the historical Jesus, both having been born in the same region of the world.