A child who became Bar Mitzvah during Shiva/Shloshim:

Some Poskim^[1] rule that a child who became Bar/Bas Mitzvah during the Shiva [or Shloshim] of a relative, is not obligated to keep Shiva or any laws of Aveilus even after he becomes Bar Mitzvah.^[2] Other Poskim^[3] rule a child who became Bar/Bas Mitzvah within the Shloshim from the burial, then he is to keep Shiva and Shloshim from that day and onwards. Practically, we rule like the former opinion that the child remains exempt from all mourning.^[4] [This applies even if he only discovered the death after he became Bar/Bas Mitzvah. If, however, he became Bar/Bas Mitzvah between the death and burial, he is obligated in Aveilus.^[5]]

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[1] Michaber 396:3; Beis Yosef 396; Tur in name of Rosh; Rosh [student of Maharam] in end of Moed Katan; Taz 396:1 [Regarding the contradictory ruling of the Taz 340:15 in name of the Derisha, that even if a child has reached the age of Chinuch he is to be educated to mourn, see Shivas Tziyon 61, brought in Pischeiy Teshuvah 396:2, that this only applies if the child has a father and is not in a Talmud Torah, in which case he is to be educated by his father to mourn. However if the child does not have a father, or he is in a Talmud Torah, then he is exempt from Aveilus, as explained in the next footnote.];

[2] The reason: As once one is exempt at the time of the obligation, he remains exempt also for the Tashlumin. [Rosh ibid] Alternatively, the reason is because the Chiyuv of Aveilus is specifically at the moment of burial, when the bitterness is greatest, and hence since the child was exempt at that time, there is no obligation for him to keep mourning laws later on. [Taz 396:1 in explanation of ruling of Michaber ibid, and answer of contradiction in Michaber ibid to his ruling in 341 where he rules like the Maharam regarding Havdalah.]

[3] Maharam Merothenberg [teacher of Rosh ,brought in Rosh in end of Moed Katan], brought in Shach 396:1 and Taz 396:1; Rabbeinu Yerucham and Bach 396 conclude to follow the Maharam, as he was the teacher of the Rosh, as well as that the ruling of the Michaber ibid is contradicted earlier in 341 where he rules like the Maharam regarding Havdalah; The Taz ibid negates his opinion; The Shach in Nekudos Hakesef defends the ruling of the Maharam against the proofs of the Taz; Yad Eliyahu 93 also negates the ruling of the Taz ibid, brought in Gilyon Maharsha 396

Within the Shiva: Some Poskim rule [unlike Michaber ibid] that if a child became Bar/Bas Mitzvah within the Shiva then he must keep Shiva for the remaining days that are left of Shiva even according to the Rosh, as every day of Shiva is a separate obligation and is not Tashlumin, and it is only in a case that he became of age after Shiva that there is a dispute. [Yad Eliyahu ibid; Poskim brought in Pnei Baruch 25:5 footnote 12 and Nitei Gavriel 127:3 footnote 5-6; Gesher Hachaim 19:3-3 that so is custom] However according to the Michaber and Tur and Taz ibid the Katan is exempt even in such a case.

After Shloshim: If he became Bar/Bas Mitzvah after the Shloshim, then according to this opinion, he is to keep Aveilus for one hour, just as is the law regarding a Shemua Rechoka. [Maharam; Pnei Baruch 25:5]

[4] Taz 396:1 “Therefore it appears Halacha Lemaaseh to rule like the Rosh, not for his reason, but for the reason I mentioned.”; Chochmas Adam 168:6; Aruch Hashulchan 396:5; Kitzur SHU”A 216:2; Ikarei Hadat 36:48

[5] Nitei Gavriel 127:4 and 8; See Taz ibid