= Solution
Use signed <velocities> in Skaro's <inertial frame>; the returning velocity $w$ must be negative. For the first two rockets, the <Lorentz transformation> of infinitesimal separations is $dx=\gamma_u(dx'+u\,dt')$, $dt=\gamma_u(dt'+u\,dx'/c^2)$. Dividing with $dx'/dt'=v'$ gives the <relativistic velocity-addition formula>
$$
\boxed{v=\frac{u+v'}{1+uv'/c^2}}.
$$
For the third rocket, its signed velocity in the second frame is $-w''$, so another application gives
$$
w=\frac{v-w''}{1-vw''/c^2},\qquad \boxed{w=\frac{u+v'-w''(1+uv'/c^2)}{1+uv'/c^2-w''(u+v')/c^2}}.
$$
Since $0<v<c$ and $0<w''<c$, the denominator $1-vw''/c^2$ is positive. The third rocket returns from positive $x$ to Skaro if and only if $w<0$, hence
$$
\boxed{w''>v=\frac{u+v'}{1+uv'/c^2}}.
$$
Equality leaves it stationary in Skaro's frame, so it never returns from $x=2L$; a smaller $w''$ leaves it travelling outward.
Let $O$ denote departure, $A$ boarding the second rocket, $B$ boarding the third, and $C$ reunion with Skaro. In coordinates $(t,x)$, their positions on the <worldline> are
$$
\boxed{O=(0,0),\quad A=(L/u,L),\quad B=(L/u+L/v,2L),\quad C=(L/u+L/v-2L/w,0).}
$$
Indeed, the first two coordinate durations are $L/u$ and $L/v$, and the returning leg has duration $-2L/w>0$. Equivalently, their $(ct,x)$ coordinates are obtained by multiplying each displayed time by $c$; these are the coordinates used for the <spacetime> diagram.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ia/paper-4-three-rockets.png]
{title=Three inertial rocket legs and reunion in Skaro's frame}
The diagram uses representative velocities satisfying the return condition; its event labels apply to the general formulas above. Skaro's elapsed time is
$$
\boxed{T=\frac Lu+\frac Lv-\frac{2L}{w}}.
$$
On a constant-velocity leg the traveller's <proper time> is $\Delta t/\gamma_V=\Delta t\sqrt{1-V^2/c^2}$, by <time dilation>. Neglecting the specified brief accelerations, Davros therefore ages by
$$
\boxed{\Delta\tau=\frac Lu\sqrt{1-\frac{u^2}{c^2}}+\frac Lv\sqrt{1-\frac{v^2}{c^2}}-\frac{2L}{w}\sqrt{1-\frac{w^2}{c^2}}.}
$$
Every contribution is positive because $w<0$. Each square-root factor is smaller than one, so $\Delta\tau<T$: Davros returns younger than a companion who remained on Skaro. The elapsed times compare the same departure and reunion events; they cannot be obtained by applying one time-dilation factor to the whole journey, whose <worldline> has three different velocities.
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