To be brutally honest, having the rate of output of ANY physical activity be essentially the rheostat setting in your system is JUST THE MOST CURIOUS THING ON EARTH.
You seem to be getting some remarkably unhelpful help.
On the other hand, the “specification” is a bit unclear. We’d really need to know what sort of “electrical signal” your pump controller needs (e.g. a voltage in some range, a current in some range, a pulse train with some characteristics).
Anyway, one common approach would be to create a pulse train whose frequency varies with cycling speed using an optical sensor or a magnetic (probably Hall effect?) sensor. Another other would be to somehow hook a small electrical generator into the “drive train”
But you’ll probably need additional stuff – which may or may not be simple – to convert the pulse train or current to a control signal with the characteristics needed by the pump controller. You really need to know what that rheostat does and what the pump controller expects to see. Without that information, you’re likely to waste a lot of time on approaches that can’t possibly work.
If you are really thinking in terms of bypassing the existing controller and controlling the 220 Volt signal to the pump directly, you’ll probably want to be looking into lamp dimmers and such. I believe that those are generally Silicon Controller Rectifiers or other mildly exotic devices. Be warned that 220Volts at 5 amperes is serious power and that on top of the hazards offered by the voltage, there is some risk that during design and test you will end up generating a very large amount of heat in a very small space resulting in some shrapnel flying around the room.
What is going through the rheostat now? Is it 220VAC or a control signal?
Okay, it’s rube goldberg but it could work, so check this out:
http://en.wikipedia.org/wiki/Liquid_rheostat
Build one of these things but put a hole in the bottom of it for draining the brine into a sump. The bike then drives another pump which moves water from the sump to the rheostat; the level of water is controlled by the brine flow into the rheostat, as controlled by the speed of the bike (pump), equilibrating with the rate of brine draining out of the rheostat, as controlled by the height of the brine in the rheostat. The height of the brine controls the power delivered to the main pump. The height/bike speed correlation will be square root rather than linear (unless you do something to make the outflow laminar like using many small tubes), but you can invert that correlation by changing the rate of change of area of the electrodes with height. Also there may be a significant time constant of output response to control setpoint (bike cadence) input. The sizes of the rheostat, the hole and the bike-driven pump are left as an exercise for the user.
This might be useful for sizing:
http://chestofbooks.com/crafts/scientific-american/sup7/Liquid-Rheostats.html
Resistive heating might be a problem but the cycling water provides an opportunity for cooling.
But if 220VAC is attached to this thing, it’s probably going to kill somebody. At a minimum the water streams going in and out of the rheostat should be broken up by a paddle to electrically isolate the rheostat from the pump and bike, and a ground-fault type of circuit to cut the 220V power should be in place.