To make notation a little bit lighter I am going to use $N=(N(t)_1,N(t)_2)$, $X=(X(t)_1,X(t)_2)$, $X'=(X'(t)_1,X'(t)_2)$.
In order to answer the question, I think we need to know what the dependency between $X$ and $N$ is?
Whatever their dependency, the only case when $Pr(X'=x') \neq 0$ is when $N=0$. If not $X'$ is continuous, because $Y$ is continuous, and $Pr(X'=x')=0$.
I hope this helps. Have you tried doing some simulations?