You could use semidefinite optimization to find that small enclosing ellipsoid.

Let $E(W,c):=\{x\mid (x-c)^TW(x-c)\leq 1\}$. Your problem is to find, given the ellipsoids $E(W_1,c_1)$ and $E(W_2, c_2)$, a positive definite matrix $A$ and a vector $z$ such that $E(W_1,c_1)\cap E(W_2, c_2)\subseteq E(A,z)$ and $Vol(E(A,z))$ as small as possible.

Minimizing the volume amounts to maximizing the concave function $\log(\det(A))$. By the positivstellensatz, the polynomial inequality $p(x):=1-(x-z)^TA(x-z)\geq 0$ holds true for all $x$ such that $q_i(x):= 1- (x-c_i)^TW_i(x-c_i)\geq 0$ for $i=1,2$ if and only if

$$p=s_1q_1+s_2q_2+t,$$
where $s_1,s_2,t$ are some polynomials that are sums of squares (SOS) (there are some technical conditions for the 'only if'). Now a polynomial $u$ of degree $2d$ is a SOS if and only if $u(x)= \tilde{x}^TU\tilde{x}$ for some positive semidefinite matrix $U$, there $\tilde{x}$ is a vector whose entries are the monomials of degree $\leq d$ in the $x_i$.

All together this gives, fixing a max. degree $d$, an optimization problem over positive semidefinite matrices $A, S_1, S_2, T$ and a vector $z$, where the entries of these matrices are restricted by linear equations that depend on the input ellipsoids. The higher $d$, the better an approximation of the optimal enclosing ellipsoid you will get. However the sizes of the SOS matrices are exponential in $d$.

Edit: Markus notes below that $p$ depends on the entries of $A, z$ in a cubic way, and I agree that that is a problem. So I guess the method above works only if we fix $z$, which is not as nice.

So here is a way out. Introduce a new variable $y$ and a new equation $y-1=0$ to the system, and put $p(x,y):=1-(x-zy)^TA(x-zy)=1-w^tBw$, where $w=(x,y)$. Then $E:=\{(x,y)\mid p(x,y)\geq 0\}$ is an ellipsoid centered at the origin when $B$ is positive semidefinite, and we can minimize the volume of $E$ as before by maximizing $\log(\det(B))$. As $E$ is constrained only to contain some stuff at $y=1$, minimizing the volume of $E$ is equivalent to minimizing the volume of $\{x\mid (x,1)\in E\}$.

To take the new equation $y=1$ into account, we optimize over all $p$ such that $$p=s_1q_1+s_2q_2+t +(y-1)u,$$
where $s_1,s_2,t$ are SOS polnomials and $u$ is any polynomial. The variables of this problem are positive semidefinite matrices $B, S_1, S_2, T$ and the free coefficients of $u$.