[Arran]

Lar;
  You could use the variac to slowly power up the Perma Power power supply to test it, but it really isn't of any use in testing battery radios unless you have the battery radio plugged into the Perm Power unit whilst you do. If it has metal can filter capaciitors can you may be able to reform or recondition them by power it up on a variac, it won't help the selenium rectifiers if they have started to wear out though, if they are still good then you might be able to use the Perma Power using it over the short term.
 Slow power ups are really only useful for reconditioning old electrolytic capacitors and for catching electrical faults in a device before they can cause any real damage, battery radios have maybe one electrolytic capacitor whilst the rest are paper and mica caps. One thing that often causes trouble in battery radios is that the wires will break down inside the battery leads right near the plugs on the ends, so you may have to cut the plugs off and strip the wires back, this you can check with an Ohm meter first.
Regards
Arran