Potentially a silly question, but do you have an example where neither $H\ne \Bbb C$ M$ nor $N$ is finite type $I$ and one of $M$ and $N$ is not infinite type $I,$ and $N\vee M'\cong N\overline{\otimes} M',$ (by the map you defined) even non-spatially (for $N$ and $M$ factors)factors? It's not clear to me that there is a non-trivial case when this actually is an isomorphism, or that there is an example where these algebras are isomorphic by a different map. For example if $N=M$ this is impossible unless $M$ is type I, since $M\vee M'=B(H).$
Also if say $M$ or $N$ is type $II$ you need to have $N$ ``far'' from $M$ for $N\vee M\cong N\overline{\otimes} M,$ for example $N'\cap M$ cannot be finite dimensional (because then $N\vee M$ is type I begin a commutant of a type $I$ von Neumann algebra). So this will rule out finite index inclusions, for example.
