the general rule with old electronics is to change first all the electrolytics capacitors, so you can have a steady power supply and make measures of the rest of the circuit clearly.
Those "tropical fishes" capacitors are prone to cracking, if you want to avoid further problems I'd suggest to replace it with a modern film capacitor.
Apart from being nice and colored, they are not better for audio than other film capacitors. If you want an upgrade use a paper in oil cap, or styroflex caps... but you probably won't hear any difference in this circuit.
Then if you still don't have any reverb effect, you may want to check the signal patch with an oscilloscope or an audio probe ( http://diy-fever.com/misc/audio-probe/
I'm not sure about the wiring of the tank, it seems only two pins are connected to the tank, the middle pin seems unconnected and is used as support for the output caps.
Yes you can replace both ceramic cap by film caps.
You need to measure the tank input and output impedance to get a proper replacement tank, it's not just a matter of spring size.
Also this effect creates the "echo" trick by chopping the reverb tail with a square wave tremolo, funny but not really an echo.
With new caps the LFO should have a better shape and you may have less volume drop with this setting.