When using Mawile GX's "Captivating Wink" Ability to put Pokemon onto my opponent's Bench, are they allowed to evolve those Pokemon on their next turn since they just came into play?
Yes, as long as it's not the opponent's first turn of the game they can evolve those Pokemon if they are able to do so. It is a basic game rule that neither player can normally evolve any Pokemon on their first turn of the game, even if they are going second.
Source: TPCi Rules Team (2020-03-12)

Relevant cards

Note: All applicable cards may not be listed below.
SM11_141
SM11_141
SM11_227
SM11_227
SM11_246
SM11_246