A workable scenario is that Taim was turned by 13 Myrdraal and held in the Blight for a while to be trained in Age of Legends weaves. I don't recall when Taim was first captured, but I recall a reference to him by some Black Ajah members some time in the early books, when they were plotting to capture and turn him.

However, assuming this happened sometime in the Great Hunt, this left VERY little time for Ishamael to train him, before he was killed by Rand.

So, for all the time that passed during tDR, tSR and FoH, another male Forsaken had to have been training Taim. Who it was remains open to speculation, but we do know it couldn't have been Aginor, Balthamel or Ishamael.

It is also unlikely that it was Asmodean, while we know that Sammael, Be'lal and Rahvin scrambled to get control of nation states in preperation for Tarmon Gaidon.

So who is left?

If Taim's mannerisms in LoC are indicative of AOL training, the timeframe is wrong for Ishamael to have been the one responsible for it.

Demandred cannot be everywhere. I have already posted theories stating that he is in Murandy, the Land of the Madmen or in Shara. He could be in none of those places for that matter. It is certainly conceivable that he didn't have time to spend on the Black Tower.

I just find it interesting that Moridin himself would personally get involved with the Black Tower, since they aren't necessarily a more important resource to the Shadow than the Seanchan or the White Tower or the Salidar Aes Sedai would be.

In any case, Moridin might well be the one controlling Taim now, but he definitely did not have the opportunity to train Taim during the period between tGH and LoC. Someone else had to have taken care of that.