Is it obligatory for the questioner to continue waking his 12-year-old brother for Fajr prayer in the mosque, despite his mother's objection, who believes he is too young and has not yet reached puberty, or should he obey her command?

There is no benefit in you abandoning waking your brother for Fajr prayer, and you are not obligated to obey your mother in her forbidding you from doing so, because obedience to parents is due when they have a valid reason for what they forbid. Your brother has exceeded ten years of age, is nearing puberty, and is keen on praying in the mosque. Parents are obligated to raise their children to adhere to the religion, for "all of you are shepherds and all of you are responsible for your flock."