Ashk'nazim do not eat meat or drink wine during what's commonly called the nine days (the first nine or so days of Av). However, they may do so at a mitzva meal, such as one that celebrates the siyum (completion) of study of a tractate of Talmud. Can an adult eat meat and drink wine at a meal in the nine days that celebrates a minor's siyum? Does it count?
Rabbi B'tzal'el Stern (B'tzel Hachochma volume 4 number 100) was asked whether a minor's siyum exempts an adult from taanis b'choros.
He cites the Rambam (Hilchos Korban Pesach 5:7) as saying that someone who became an adult between Pesach and Pesach sheni need not offer the korban pesach sheni. The explanation of this Rambam is that, because the Torah says to include an entire household, including children, on the first pesach offering, it's as though the then-child fulfilled the mitzva just like an adult (even though normally we say children are exempt from mitzvos).1
Rabbi Stern extends this to Torah study: because a parent has a Torah obligation to teach his son Torah, it's as if the son has fulfilled a mitzva by studying Torah just like an adult. Therefore, Rabbi Stern concludes, his siyum can be used to exempt an adult from taanis b'choros.
(Rabbi Stern's t'shuva is cited as practical halacha by Rabbi Simcha Bunim Cohen.)
It seems to me that the same logic applies to eating meat or drinking wine during the nine days. The Torah study is like an adult's, the siyum renders the meal a mitzva meal, and those participating in the siyum can eat meat and drink wine. But consult your rabbi for a practical ruling.
1 This follows Rabbi Stern's understanding of the Kesef Mishneh (ad loc., citing Rabbi Yosef Kurkus) and Mabit (קרית ספר, ad loc.), that the father's obligation to include his minor son in the korban pesach essentially allows an adult-level fulfillment of the obligation by the son.
An alternative understanding of the Kesef Mishneh is expounded by Rabbi Chaim Soloveitchik. He argues that there are two legal aspects to the fulfillment of the korban pesach commandment. The first is the individual's obligation to perform a commandment, and the second is the legal completion of the sacrificial service. Rabbi Soloveitchik contends that only the second aspect of the korban pesach applies to a minor, so the minor's parent is capable of including him in a household group for the korban's consumption. Since the korban pesach service could be performed by a minor, its prior performance has legal significance for him once he reaches the age of majority. However, it is not as if he actually fulfilled an obligation as a minor.
Rav Eliyashiv (Ashrei Ha'ish 69:12) held that a katan who makes a siyum and makes a celebratory meal is not considered a seudas mitzvah which would permit one to have meat and wine.