Scholars from all the major religions are down with alien life. Muslims, Christians. Hindus, Jews, and Buddhists wouldn't have a problem with them. Neither would Jains, Sikhs, Rastafarians, Pastafarians, Dischordians, Baha'i, Satanists, Voduns, Wicians, Druids, Asatrus, Caodaists, Santerians, Yazdânists, Zoroastrians, Taoists, Confusionists, Shinto, Damanhurs, or Ifans.
The way I see it, even the enlightened and advanced culture of TOS would be running so low on food, fuel, supplies, and space by the time the arrived at Earth that they wouldn't be able to wait. They would have no choice but to dive in head first. Furthermore, they will know that it is only a matter of time before the Cylons find them again, meaning that they must help Earth build a defensive fleet as quickly as is possible, or else all may be lost.
These twin concerns mean that they must make contact immediately, though they would do so in a manner that produces the least disruption. Smart aliens would make radio and visual contact with the ISS first, that pretty much ensured that people won't think it is a hoax, and allows them to set up contact with people in power without relying on a broadband transmission that would be misinterpreted. The important thing would be negotiating for a landing site for the refugees, with the second issue being meetings with world leaders to discuss technology exchange. Galactica would stay back, probably in the Asteroid Belt, both to avoid detection by the Cylons if they ever arrive and so that the Battlestar's presence will not be seen as hostile. Adama would personally meet with the G8+5 to discuss the issue of preparing for the defense of Earth just in case the Cylons find them again.
After a period of quarantine, the refugees would quietly assimilate into the Albuquerque, New Mexico metropolitan area, or some some place similar. They would most likely maintain close ties and have their own little community, for a while, but these happy-go-lucky fellows would try their best to become a part of Earth society. It would be a lot like Alien Nation, without the spots or the men giving birth.
Adama, not knowing anything about Earth politics but having common sense, would not give a military advantage to any one country alone. Rather, he would demand assurances that the technology he shares will be used in the defense of Earth, not the domination of it. Luckily, there are already international mechanisms in place to ensure that. China, Russia, and a few other countries would join NATO, forcing them to drop the N and the A. Each country would get as many battlestars as they could afford, but all space defense personel and equipment would be placed under the command of a permanent special TO task force. Being outside the normal chain of command, no one country could use it for political posturing. Once that agreement gets hammered out, Adama will send engineers from Galactica to Earth to help them start building ships.
Meanwhile, Richard Branson will purchase a couple of older civilian ships that have been decommissioned. One he reverse engineers, the other he turns into his personal pleasure craft. As the TO countries build their battlestars Virgin Spaceways builds it's own commercial vessels that make civilian colonization of the stars economically feasible. That, combined with natural human desire for frontier exploration and the Eggs in One Basket problem exacorbated by the existance of the Cylons will lead to cities on the Moon and Mars, mines in the Asteriod field, and eventually colonization of the far solar system and beyond.
The space race would be on again, but this time the competitors would be corporations. There would be a strong temptation to give those corporations government sponsorship, as with many exploratory trading companies during the most recent age of colonization, but common sense would win out and fair treaties about the exploitation of space resources would be signed, treaties that would severely limit the activities of major corporations, leading them to curtail their interstellar exploration for a while. This wouldn't be a bad thing, because more exploration and colonization also means a bigger target for the Cylons to find.
On Earth, not much would change really, in the end. The more weird shit happens, the more things stay the same.