What is the age in which hypogonadism in males needs to be taken seriously?

Treated after age 13. Usually boys aren't diagnosed with delayed puberty until they're at least 13. By the time they're 14, with little or no pubertal development, some may need a "jump start" with testosterone. It takes sometimes years to know if a boy is simply delayed or truly hypogonadal unless he has testicular failure.
Now. There are various causes of hypogonadism, but they all should be followed by a specialist.