You're looking for the projective tensor norm. Explicitly, for $\omega$ in the algebraic tensor product $X \otimes_{\mathbb{R}} Y$ it is given by \[ \Vert \omega\Vert_{\pi} = \inf\left\{\sum \|x_{i}\|\,\|y_{i}\|\,:\,\omega = \sum_{i=1}^{n} x_{i} \otimes y_{i}\right\} \] and the associated tensor product $X \hat{\otimes}_{\pi} Y$ is the completion of the algebraic tensor product with respect to that norm (the above expression obviously defines a semi-norm, in order to show that it is a norm, you have to think a little bit).
Since $X \otimes_{\mathbb{R}} Y$ sits inside $X \hat{\otimes}_{\pi} Y$ you get a bilinear map $m:X \times Y \to X \hat{\otimes}_{\pi} Y$ by sending $(x,y)$ to $x \otimes y$ and by definition $\Vert x \otimes y\Vert_{\pi} \leq \|x\|\,\|y\|$ so that $\|m\| \leq 1$.
The basic thing you have to check is that the above inequality is in fact an equality $\Vert x \otimes y\Vert_{\pi} = \|x\|\,\|y\|$ and using this it is easy to prove that a continuous bilinear map $\Phi: X \times Y \to Z$ yields a unique linear map $\varphi: X \hat{\otimes}_{\pi} Y \to Z$ of the same norm (by the property of the algebraic tensor product $\Phi$ gives a linear map on $X \otimes_{\mathbb{R}} Y$ and if you've proven that this linear map is continuous, it is continuous on a dense subspace, hence extends uniquely to $X \hat{\otimes}_{\pi} Y$).
In other words the map $m: X \times Y \to X \hat{\otimes}_{\pi} Y$ yields an isometric isomorphism $\displaystyle B(X \hat{\otimes}_{\pi} Y, Z) \to B^2(X,Y;Z) $ by pre-composition $\varphi \mapsto \varphi \circ m = \Phi$. As you already observed $B^2(X,Y;Z) \cong B(X,B(Y,Z))$ which yields the isometric isomorphisms \[ B(X \hat{\otimes}_{\pi} Y, Z) \cong B(X,B(Y,Z)) \cong B^2(X,Y;Z) \] as in the case of vector spaces. In fact, you can convince yourself that you have to define the projective tensor norm as above if you want the first isomorphism to hold, already for $Z = \mathbb{R}$.
You can find all this and much more in great detail in R.A. Ryan's book Introduction to tensor products of Banach spaces, Springer 2001.