How long does it take to feel the effects of testosterone replacement therapy for a teenager with hypogonadism?

Depends on levels. I usually see improvement by the second or third injection and that tends to correlate largely on serum levels. Depot injection tend to have peak/trough effect versus gels which provide a more "physiological" absorption rate, however my experience with gels in teen age boys have disappointing to say the least ! (added the risk of skin transmission to other children).
Pretty soon. Usually does not take long but we like to start the treatment slowly mimic normal pubertal development that in a boy can take 2-4 years.
1-3 months. Raising one's testosterone level has effects throughout the body. The endocrinologist who is prescribing the testosterone will have the most accurate answer to "how long before a patient notices improvement". Probably the improvements in mood and sexual interest will come first, maybe in a month. The changes in skin, hair, penis, testicles, muscles, blood, etc., will probably come in 3 months.