Since we're not having to deal with friction, conservation of energy is our best friend. There's no need to "convert rotational work into translational work", merely observe that they must be equal in a stable object. So
$$W_F = W_T$$
where $W_F$ is the work done by the applied forces and $W_T$, by the torque.
A note before we continue: a screw's pitch is usually defined as the axial distance between two threads. However, it can actually be given in two different ways: metric screws usually give you the pitch as a distance (i.e. 1.5 mm), while inch-based screws tend to be given in terms of threads-per-inch.
For the derivation below I'll be using pitch as a distance.
So, if we think of a single rotation of the screw, the plate will move along the rod by a length equal to the screw's pitch $d$ (the linear distance between two threads).* Therefore, we know that
$$W_F = Fd$$
As for the rotational work, we can imagine it is created by a force $f$ applied tangent to the nut, rotating the screw once around the rod's spiral. We can save ourselves some work and consider the length of that spiral equal to the circumference of the rod.
$$\begin{align}
W_T &= 2\pi rf \\
&= 2\pi T
\end{align}$$
The last steps are now trivial:
$$\begin{align}
W_F &= W_T \\
Fd &= 2\pi T \\
\therefore T &= \dfrac{Fd}{2\pi}
\end{align}$$
If you instead have the pitch defined in threads-per-distance, the modification is trivial:
$$T = \dfrac{F}{2\pi d}$$
where the torque's distance unit is the distance unit in the pitch's threads-per-distance (usually threads-per-inch, therefore inches).
* Strictly speaking, the dimension $d$ to be used is the lead, not the pitch. The lead is the axial distance covered by one rotation around the screw, while the pitch is the distance between two threads. In most screws, these values will be the same (as in the left side of the image above) and pitch is the more common term, which is why I used it above. However, when using multi-start screws, the lead will be greater than the pitch (right side of the image), and $d$ should be adopted as the lead.
