Assalamu alaykum wa rahmatuLlah,
Masha'Allah, that's quite a story. Alhumdulillah on your acceptance of Islam, that's the most important point I think, first and foremost.
Right, with regards to your situation, there are two distinct issues as far as I can see:
1 - the Islamic fiqh perspective
2 - the effect on society, etc
Let me deal with the fiqh perspective first, because that's dead easy.
The nikah is a very simple process, and requires 5 people as a minimum, the bride and groom, the wali, and two witnesses. The groom and wali need to make a statement offering and accepting marriage, without any ambiguity or delay, for the marriage to be established (it's implied the bride has given her permission). The two witnesses are there for verification that it happened, but it's good to get a proper certificate drawn up too.
There doesn't need to be any maulana's, sheikhs, imaams, or whatever present. People like to include them to make sure everything is being done properly, and perhaps to recite some surahs of the Qur'an and make dua, but it's not a requirement for the establishment of the nikah.
However, once the nikah has been established, and you've lived together and consummated the marriage, a walima is wajib/obligatory. This is a party to announce the marriage has happened, and it has many rulings and benefits that can be discussed later insha'Allah if you get to that point.
My own opinion though, having been in a similar situation as to yourself, is to be very careful. I've been married twice myself, and in both cases the families weren't happy about it, but gave into the girl's wishes (well, the first one did, the second one got a little extreme).
My point is that, even though they permitted the marriage, as soon as problems came up, they didn't work to maintain the marriage, but instead to destroy it. They kept planting thoughts and ideas into my wife's mind, and kept giving her tasks that worked against our marriage, and after several years, they succeeded. They would rather their daughter be divorced and the stigma attached to that, than to be happily married to someone they didn't pick.
I'm not saying it can't work for you, just to be careful. The girl needs to be strong. I'd also advise you to pray salat-al-istikhara, but one of the conditions for it is neutrality, and given that both of you have already decided on a course of action, I don't think it'll help you much. But it's up to you.
In short, the nikah isn't an issue, it can be done very easily. You can appoint your uncle as an agent to perform it on your behalf. Your difficulty will come to the walima, people may ask questions as to your background. You need to decide as a family how you're going to answer this. I've seen people get around it, so it can be done, but you just need to decide between the family what your response is going to be, or you could even just dismiss the question if asked because it has no relevance on anything.
May Allah grant you what is best