Firstly: 

One of the conditions of marriage being valid is the presence of a wali; it is not permissible for a woman to give herself in marriage without a wali. If she does get married without a wali, then her marriage is invalid. 

There is no guardianship of a disbeliever over a Muslim woman in any circumstances, according to scholarly consensus. 

Please see the answer to question no. 48992. 

Secondly: 

Guardianship in the case of marriage belongs to the ‘usbah or male relatives on the father’s side; this includes the father, the (paternal) grandfather, the son, the brother and the paternal uncle. 

With regard to relatives on the mother’s side, they are not ‘usbah, and they cannot be guardians in the case of marriage. The maternal uncle does not come under the heading of ‘usbah, so he cannot be a guardian in the case of marriage. 

But there are some cases in which the maternal uncle may act as the guardian in the case of doing the marriage contract. These cases are: 

1.     When the wali has appointed him to do that on his behalf.

2.     When the marriage contract has already been done, and it has been officially documented in a Muslim country that follows the scholarly view that it is permissible for a maternal uncle to give his sister’s daughter in marriage, or the view that it is permissible for a woman to get married without a wali (guardian).

Please see questions no. 152595, 153602

Thirdly: 

If the woman does not have any wali who can give her in marriage, then that role may be filled by the Muslim qaadi (judge), if there is one. If there is no qaadi, as is the case for Muslim communities living in the West, then the director of the Islamic centre may give her in marriage. 

If there is no director of an Islamic centre, then the imam of the mosque, or a scholar, or a Muslim man of good character may give her in marriage. 

Shaykh Ibn Baaz (may Allah have mercy on him) said: 

If the woman is in a country where there is no wali – no brother, no father, and no paternal cousin – then the ruler takes the place of the wali; her wali is the ruler, because the Prophet (blessings and peace of Allah be upon him) said: “The ruler is the wali of the one who does not have a wali.” So the ruler takes the place of her wali and becomes her wali; he may give her in marriage or appoint someone else to give her in marriage. If she is in a land where there is no (Muslim) ruler, qaadi or wali, as in the case of Muslim minorities in non-Muslim countries, then the director of the Islamic Centre may give her in marriage, if there is an Islamic centre, because his position is like that of the ruler for them. The director of the Islamic Centre may look at her case and give her in marriage to one who is compatible, if she does not have any guardians of her own and there is no qaadi. If the wali lives far away, he should be corresponded with ot spoken to until he appoints someone as his proxy (wakeel). But if it is not known where he is, then the next closest wali takes his place; the one who is most closely related to him takes his place. If she has no wali except this absent one whose whereabouts is unknown, then the ruler takes his place. 

http://audio.islamweb.net/audio/index.php?page=FullContent&audioid=290408 

In conclusion: 

The director of the Islamic centre in that country may take the place of the ruler, and the guardianship of the ruler in the case of marriage is an Islamically-acceptable kind of guardianship, if there is no wali who is a male relative on the father’s side (‘usbah). You do not need to travel to your maternal uncle in order for him to give you in marriage himself. 

If this man becomes Muslim because he genuinely wants to follow the religion, and his Islam is real, then there is nothing wrong with the marriage contract being done by the director of the Islamic centre in the place where you live. 

And Allah knows best.