Local co-op will not be available on the New 3DS version of The Binding of Isaac: Rebirth, confirms Tyrone Rodriguez, the founder of Nicalis. Tyrone also notes that patching it in after release is unlikely. Cross save is also not available between the New 3DS and Wii U versions.
The upcoming Wii U and Xbox One ports of Rebirth will all have local co-op intact, just like the recently released PlayStation 4 version. No release date has been announced just yet, but I have a sneaking suspicion we will find out at E3 as Tyrone noted, "June's a good time to announce the release date, but waiting on a few things before we can do that."