The effect on Crohn's is determined by a controlled suppression of the immune system and depends on how long white blood cells on average live (several weeks) in the particular human AND on that individual's reaction to a certain dosage of azathioprine (imuran) as well as how regularly the individual takes the drug (e.g. non-conformance of patient, that is disregarding when and how to take a dosage). It usually, as said above, takes 2-3 months.
The effect is not a "switch" in that after a certain time, the person just feels good. Azathioprine only helps a patient to manage Crohn's, but a detrimental way of living (eating in a way detrimental to Crohn's, heavy alcohol consumption, tobacco use, no sport, stress etc.) will in most cases "overpower" the useful effect of azathioprine. These patients usually report that azathioprine did not help them stay in remission. Also, azathioprine has not been determined to have any statistically significnt benefit to induce remission in patients with active Crohn's (that is why steroids are required; or biologics are used to induce remission).
Regarding "leaving the system", azathioprine is to an extent of 90% absurbed directly by the gut. The highest blood plasma concentrations are one to two hours after taking it, depending on the patient and the average plasma half-life (once it reaches the blood) is 20 to 80 minutes for azathioprine and 3 to 5 hours for its metabolites (principally 6-mp). 20 to 30% are bound to plasma proteins while circulating in the bloodstream. Having said all that, after 3 times the half life and considering the original intake, after 12 hours, most orally taken azathioprine has left the system and after 24 hours, the system as returned principally to normal. This is also the reason, why aza should be taken in the morning and the evening and not just one time per day, each time at the same dosage.