All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Yes, it is permissible for a fasting person to cut his nails and trim his mustache and do other Fitrah acts (acts of natural disposition) as Allaah has not prohibited them. it is only the pilgrim or the person who is performing Umrah who are prohibited from doing so and the jurists mentioned these among the prohibitions when in a state of Ihraam.
It is also prohibited in Islamic legislation to have a hair cut or cut one's nails during the first ten days of Thul-Hijjah for a person who wants to offer a sacrifice, as per the Prophetic narration reported by Imaam Muslim.
Allaah Knows best.