Mechanism to draw rear wheel back and tighten chain stuck; oil not helping, what's next?
I'm a little puzzled by the threads that look like they should draw the hub's axle back to tension the chain. They don't appear like they can move, but it also doesn't look like there is a nut on them to pull the axel back. Do you know if they thread into the frame, or are they free to slide?

That said, it appears that the load on the mechanism would tend to misalign the threaded part where it passes through the frame. This is very likely to cause it to jam. I think you could try very gently prying up on the loop where the axle attaches or perhaps tapping down gently on the top of the loop. It shouldn't take much force at all. I'm assuming that the frame is not threaded and that the adjuster should slide freely.