My question is:
Consider an alternator producing 20 MW power, while consuming 20 MVAR i.e being under-excited. What would be the instantaneous current flowing? Since the direction of VARs is opposite to that of MW, will their currents cancel out each other?
Reactive power and real power will not "cancel each other out". The combination of reactive power and real power is the apparent power, which is expressed as a complex number. The real part represents the real power while the imaginary part represents the reactive power. I would suggest you read through the Wikipedia article on AC power for a good high level explanation.
In your specific situation you have 20 MW of real power and 20 MVAR of reactive power, this is equivalent to 20*2 = 28.3 VA of apparent power. You did not specify the voltage in your post so it is impossible to say what the current in the wire will be.