I'm going to make this short itself as I'm working on a paper but think of the NAU this way

The United States will be one part as a whole, all 50 states combined will make up 1 / 3rd of the union. these states dont become separate entities. Its not like the EU separated their countries states (although there isnt a good example for sake of argument) but they allowed each country to join as a separate entity to the Union. For example when Sweden joined the union they did not lose the title of Sweden or any "states" if they had any.. What i'm getting at is primarily this quote
"The European Union is composed of 27 independent sovereign countries"
The NAU would not remove the states from America, it would keep them intact, much like the provinces in canada would be intact. There is little separation of the countries intact.

The "American way" will still thrive in the United States.

"For example, if Muslims ruled where Westerners lived and that was our main ideology basis for our laws, we would be marrying 6 year old girls who call themselves Muslim."
I take this quote from your statement and it is true, if that were actually the way the EU system was run than you would be correct in the outcome, though the system in which the union is run through a supreme council etc prevents one person / belief to run the Union. Much like the house of commons in canada where all political parties have the ability to earn seats and have their voices heard in parliament.

The religious stuff I will not get into as I do not really follow nor participate in much of it at all.

Overall the NAU (if it would ever occur) and yes I am doubting the occurance due to the gain and loss of each nation. overall the NAU would not separate the united "states" it would combine what the United States brought to the Union as one entity, or 1/3rd of the Union. Canada's provinces would be 1/3rd of the union and Mexico would be 1/3rd of the union. Although proper weighting towards the council / supreme powers of this union would have to be taken into account as mexico probably would not get as much representation compared to Canada or the United States.
Thats just how I see the union.