Braking cages was discussed some time ago in regional topic ( Игровая вопросница, in Russian), here are some thoughts from me (but note that I was assuming character is already armed and equipped):
If the character is very strong (12+), then you can take a sledgehammer and hit on resin gratings, it should break in 10-15 hits. It will be faster than chopping for 1 hour with a pickaxe through the action menu. If the character is very strong, then, with some luck, he/she can break cage from the first blow, I think.
I managed to free the NPC like that. But releasing prisoners from their cages is very easy compared to making these fools go outside. They became insane because of the heat, and they will run at random and not follow you, and they can only come to the exit from the tower by accident. By some miracle, I managed to free the prisoners in one of the towers and lead them out.
As for the heat - I was playing as a experienced character, with “Internal climate control”, “Heat dissipation” and “Repair nanobots” CBM modules, and was able to operate inside the tower in a gas mask for a relatively long time, before getting health dropped (but not for hours, of course)
Another stupid plan (you can try and tell us) - go to the tower outside, bury yourself with a pickaxe under the ground, dig a tunnel underground under the prisoners’ cells (count the tiles), and then try to dig up with a pickaxe. Or put explosives under chamber C4 - if you’re lucky, the explosives will break the floor and the prisoner will fall into the tunnel.