Is it allowed in any circumstance to take a haircut on rosh chodesh?
Under difficult and extenuating circumstances, it is permitted to have a haircut on Rosh Chodesh.
Refraining from taking haircuts on Rosh Chodesh is a matter of custom, as mentioned by the Magen Avraham (260), which is based on the tzava’ah of Rabbi Yehudah Ha-Chasid.
Poskim write that those who are particular to refraim from haircuts on Rosh Chodesh should do so even on Erev Shabbos (despite the mitvzah of taking a haircut before Shabbos).
However, the Mishnah Berurah mentions the custom of some to have a haircut on Rosh Chodesh Iyar and Sivan, and this is mentioned by the Radvaz (687).
Under difficult and extenuating circumstances, it is permitted to have a haircut on Rosh Chodesh, even for those who generally refrain from doing so. The general refrain from this will usually not be considered a ‘neder,’ because people don’t see it as a voluntary act. Even if we see it as a neder, if a person did not refrain from taking a haircut under similar difficult circumstances, it follows that the neder was not accepted for such circumstances (Dagul Mervavah, Yoreh De’ah 214).