Is it carried away as momentum imparted on the [product] atom? Is it carried away in neutrinos? Is it carried away as gamma rays?
All of these can happen, and in general nuclear reactions will output their energy via a combination of these. The specific combination, of course, depends on the specific reaction.
Also, if neutrinos are massless, can they carry energy away as momentum, or is that not possible?
As it happens, neutrinos are not massless (or at least some of them are not; the jury is still out on that but the situation is slightly more complicated). For the purposes of most nuclear reactions, this mass is small enough that you can neglect it, but this does not pose a problem. In general the energy of a particle with mass $m\geq0$ and momentum $p$ is equal to
If $m=0$, or it is very small, then the particle can still carry momentum and an associated energy. In general, neutrinos can carry away a significant amount of energy via this mechanism.