Interesting question, however it is better suited for math (or physics) stackexchanges. It also needs a rewrite with LaTeX. Here is an incomplete answer.
From what you stated it seems the two ellipsoids are similar. Let's say $A>B>0$ so ellipsoid $B$ is inside $A$ at the start. Also assume $a\ge b\ge c >0$. Note that the vector from center of $A$ to center of $B$ is $D=(V_B-V_A)t$.
Let's first simplify the problem and say the ellipsoids spin randomly as opposed to spin around fixed axes. Then as soon as $|D|$ becomes equal to sum of longest axes of the ellipsoids they must intersect (if they have not intersected already), that is $|D|=a/\sqrt A + a/\sqrt B$. This gives an overly simplified indication for the collision time, $t_c \le (a/\sqrt A + a/\sqrt B)/|V_B-V_A|$.
Perhaps you meant something like this
Two ellipsoids at time $t=0$ are at $(x-A_1)^2/a^2+(y-B_1)^2/b^2+(z-C_1)^2/c^2=1$ and $(x-A_2)^2/a^2+(y-B_2)^2/b^2+(z-C_2)^2/c^2=1$. They start to spin at a very high rate, $\omega=\infty$, around certain axes through their centers $(L_1,M_1,N_1)$ and $(L_2,M_2,N_2)$, while at the same time their centers move uniformly towards the origin at speeds $S_1$ and $S_2$ . I want an estimate for the time to collision (if there is one).