It all depends on what juicy spoilers about each bastard's future role GRMM gave them to work out this merger of the two characters, and if they intend to still keep Gendry in the series on an off or after his "rescue" he'll return only when he meets Brienne. It seems likely one of the two character's ancestry won't be relevant, or else it's workable to leave one arc out completely.

All that seems obvious for now is that in early s4 Gendry will be left on Dragonstone in Davos' care as Stannis and Melisandre go north, to be sent after when Melisandre needs him.

That will leave Gendry time to appear in a few scenes with Davos and with his cousin (in case he's supposed to develop a bit the relationship Shireen is supposed to have with Edric), and eventually in s4 when Melisandre wants him sent to the Wall, or before she does, Davos will decide to organize his escape, with or without escort.

They could easily send him to the Inn then, or to Lys. He can even do both.. first escape to go hide as a smith at the Inn, then later be forced to cross to the Free Cities after he's met Brienne.

They could even decide after meeting Brienne he will bump into Arya and cross with her to Braavos, or continue to Lys. He could provide a known face for Arya to interact with while in Braavos.

I suspect Edric's future problems in Lys might either involve the red temple there, or he could be there to get involved with Tyrion, Dany and co. eventually (he looks just like Robert...). Of course perhaps GRRM will bring Edric back to Westeros and Melisandre - or he will one way or another die, and it's why it's so easy to merge him with Gendry.

But it seems somewhat likely to me that Gendry isn't done with Brienne and might even end up traveling with her in WOW, in which case he'll play Edric's part and be sent in hiding... to retake his Gendry's role.

We'll see.