In order for the marriage contract to be valid, it is stipulated that it should be done by the woman’s wali (guardian) or wakeel (proxy), because of the hadeeth in which the Prophet (blessings and peace of Allah be upon him) said: “There is no marriage except with a wali.”
Narrated by Abu Dawood, 2085; at-Tirmidhi, 1101; Ibn Maajah 1881 – from the hadeeth of Abu Moosa al-Ash‘ari; classed as saheeh by al-Albaani in Saheeh at-Tirmidhi. He also said: “Any woman who gets married without the permission of her wali, her marriage is invalid, her marriage is invalid, her marriage is invalid.”
Narrated by Ahmad, 24417; Abu Dawood, 2083; at-Tirmidhi, 1102; classed as saheeh by al-Albaani in Saheeh al-Jaami‘, 2709
The wali of the woman is her father, then his father (paternal grandfather), then her son, then his son (this is if she has children), then her brother through her father and mother (full brother); then her brother through her father only (half brother), then their (brothers’) sons, then the paternal uncles, then their sons, then the paternal uncles of the father, then the ruler.
See al-Mughni, 9/355
Based on that, it is not valid for him to be a wali for marriage, because the kaafir cannot be a wali in the marriage of a Muslim woman, according to scholarly consensus.
Ibn Qudaamah (may Allah have mercy on him) said: With regard to the kaafir, he has no wilaayah (guardianship) over a Muslim woman under any circumstances, according to the consensus of the scholars, including Maalik, ash-Shaafa‘i, Abu ‘Ubayd and ashaab ar-ra’y. Ibn al-Mundhir said: All of the scholars from whom we acquired knowledge are unanimously agreed on that.
End quote from al-Mughni, 9/377
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: If he does not pray, it is not permissible for him to do the marriage contract for any of his daughters. If he does so, then the marriage contract is invalid, because one of the conditions of being the wali (guardian) of a Muslim woman is that he should be a Muslim.
End quote from Fataawa Noor ‘ala ad-Darb
If her brother was an adult of sound mind, then he is her wali. If he appointed her maternal uncle to do the marriage contract, then the marriage is valid. However the maternal uncle is not one of the ‘asbah (male relatives on the father’s side), so he cannot be the wali in the case of marriage and his giving the woman in marriage is not valid except in two cases:
(i) If the wali appointed him as his representative. If he did not appoint him, but he was present when the marriage contract was done and he approved of it, then perhaps this takes the place of (explicit) appointment and permission, because silence in such a situation constitutes approval.
(ii) if the marriage contract has been done and was officially documented in a Muslim country that adopts the [scholarly] view that it is permissible for the maternal uncle to give his sister’s daughter in marriage, or the view that it is permissible to do the marriage contract without a wali, then the marriage contract is valid, because the ruling of the judge in matters that are subject to differences of scholarly opinion is to be implemented and not annulled.
And Allah knows best.