If $\mu$ and $\nu$ are, respectively, prosection valued measures on $X$ and $Y$, then $\mu \otimes \nu$ is a projection valued measure on $X\times Y$. I don't see anything more than that going on here.

In answer to the general question about operator powers, it looks like what is being done here is that we have two commuting normal operators ($1\otimes u$ and $\hat{N}\otimes 1$), so they can be simultaneously realized as, say, multiplication by $f$ and $g$, and the power is taken to be multiplication by $f^g$, which is well-defined if $g$ is integer-valued and $f$ doesn't vanish.

If $\mu$ and $\nu$ are, respectively, projection valued measures on $X$ and $Y$, taking values in $B(H)$, then $\mu \otimes \nu$ can be defined to be a projection valued measure on $X\times Y$ taking values in $B(H)\otimes B(H)$. I don't see anything more than that going on here.

In answer to the general question about operator powers, it looks like what is being done here is that we have two commuting normal operators ($1\otimes u$ and $\hat{N}\otimes 1$), so they can be simultaneously realized as, say, multiplication by $f$ and $g$, and the power is taken to be multiplication by $f^g$, which is well-defined if $g$ is integer-valued and $f$ doesn't vanish.