Many earlier Poskim deal with this question. As for the psak halacha:
Reb Shlomo Zalman Auerbach zt”l says it is only in the second Adar. (Hlichos Shlomo in footnote)
Reb Yitzchok Zilberstein (Chashukei Chemed on Meseches Taanis) quotes his father-in-law Reb Elyashiv zt”l also as leaning to say it is only in the seocnd Adar.
Reb Shmuel Vozner st”l in Shevet Halevi (Volune 10) also says it is only the second Adar, athough he says some are noheg in the first Adar too.
Reb Chaim Kanievsky shlita- Some quote him as saying Mishenichnas Adar is also in the first Adar. However, Rav Dardak in his Sefer Yismach Yisroel says that Reb Chaim told him only in the second Adar. When asked about the others who quote Reb Chaim differently, he said that when he wrote his Sefer, he asked Reb Chaim what to write l’hahalcha l’maaseh, and Reb Chaim responded only in the second Adar.
As an aside, most poskim do not make a big deal about the simcha which is included in this mitzvah, even in the second Adar. First of all, it is not brought at all in Shulchan Aruch. The Gemara only says that it is nogea if one has a din with an akum, he should try to push it off until Adar. Reb Shlomo Zalman says that one should remove worries and sadness from his heart during Adar. While Reb Chaim shlita also does not make a big issue out of the simcha this halacha mandates.