The second hand tenor was in worse shape than I thought. Some of the rods that drive the keys were bent and the complicated mechanism that works the octave keys was seized up. There were loads of slight leaks around all the pads so that a lot of notes, even the ones not affected by the bent rods, wouldn't play - well they made some sort of sound, but wouldn't play properly and in tune.
I figured I hadn't got much to lose, so I took the whole instrument apart down to the last screw and washed all the parts in the bath! Then I gently bent the rods back as straight as I could and put the whole thing back together again, cleaning, oiling and adjusting all the pivots as I went. I made a light out of some of those LED strip lights that I could drop down inside the sax in a darkened room - then with all the pads held closed I could spot where the light (therefore air too) was leaking out.
Then a whole lot of adjustment: tweaking adjustment screws; glueing on new bits of cork to regulate the height of the pads and buttons; bending the pad cups so they sealed better on the tone holes; re-corking the neck where the mouthpiece fits on. I also had to drill and ream out part of the octave key mechanism!
Anyway, now it plays. It doesn't sound as clear as my alto, but that may be down to the way I'm playing it. I think it will need a new set of pads fairly soon, but it is working right now.
I'll take it along to my lesson next week instead of the alto and see what my teacher thinks of it.