Assume he is performing hajj tamattu and intends to sacrifice animal during hajj.
For this exact situation there's a clear ruling based on the hadith of Umm Salamah:
When any one of you intending to sacrifice the animal enters in the month (of Dhu'l-Hijja) he should not get his hair or nails touched (cut).
It was said to Sufyan that some of the (scholars) did not deem this hadith to be Maffu'. He said: But I deem it as Marfu' (i. e. chain of narration traceable right up to the Holy Prophet). (Sahih Muslim and other hadith compilations)
He who has a sacrificial animal with him whom (he intends) to offer as sacrifice, and he enters the month of Dhu'I-Hijja, he should not get his hair cut or nails trimmed until he has sacrificed the animal. (Sahih Muslim and Sunan abi Dawod)
this ruling applies especially for the case of tamat'u and all pilgrims who intend to sacrifice and non-pilgrims who intend to offer a sacrifice for 'Id al-Adha. It is haram to cut his nails or hair after wearing ihram (which can basically be done until the day of 'Arafah). Some scholars regard it as sunnah to cut nails and hair when one wants to wear ihram (before wearing it).
As for cutting nails and/or hairs for those pilgrims who intend to offer a sacrifice there are some further rulings if the first ten days of dhil-hijjah have started: The shafi'is and the majority consider it as makrooh to cut hairs or nails even before wearing ihram once dhil-Hijja has started while the hanbalis even consider it haram. But if one has done so there's consensus that he must repent but there's no prescribed fidya for doing it.