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This is always true (without nuclearity): If $T_j:E_j\to F_j$ are continuous linear maps between Hausdorff locally convex spaces and $E_2$ is complete then $$ T_1\hat\otimes_\varepsilon T_2: E_1 \hat\otimes_\varepsilon E_2 \to F_1\hat\otimes_\varepsilon F_2$$ is injective if so are $T_1$ and $T_2$. This is 16.2.2 in Jarchos's book Locally Convex Spaces.