I seem to remember finishing the UN after 2050 and being allowed to hold a vote. I don't know how it will work if you take over the city with the UN in it.
Also a factor, before the patch, if I built the UN and didn't take a vote that turn, the computer would come back every other turn and ask for a vote. After the patch, if I turn down the vote the first time, they never ask again. I waited 11 turns once and never got asked again. I had to reload to the year I built the UN and take the vote then to get a diplomatic victory before time ran out.
As far as getting others to vote for you, I always give away tech the turn before the vote. I try to get all the other civs to be gracious towards me or at the least, polite. The other consideration is, that whoever else is up for the vote, meaning they're running against you, will NEVER vote for you. They will always vote for themselves. Bribery is the key, although if you've wronged a civ in the game, they tend to hold it against you to the end and may abstain rather than vote against you if you bribe them enough.
Hope that helped.
