This is mostly related to US "normal" day to day usage of the spelling alphabet. I am new to the country and most often emails/names etc needs to be spelled and I find it difficult to determine which set of words to stick to. The Alpha,Bravo..(NATO) alphabet sounds too unnatural to me like in some hollywood film. I 've also heard people using common names (Alice,Bob..) Is that the commonly used set for everyday situations like calling the customer care centers and making reservations?
It really doesn't matter much what spelling alphabet you stick to, unless you are working in a specific branch of industry where people expect a specific alphabet.
In general, people just want to know if you mean N or M, and whether you make that clear by saying Nancy or November, your message will be clear.
In situations where a specific alphabet is used often, it does pay off to stick to it, since people expect to hear specific words when you are spelling. The NATO alphabet is not at all strange, uncommon or unnatural, for instance. It is not only used in the military, but also in civil aviation.
I do understand the "I seem to be in a movie"-feel when you start using it, and outside the military or (civil) aviation there are indeed many places where people will assume you watch too many movies when you spell square as sierra quebec uniform alpha romeo echo. On the other hand, context is everything. In my current job people will assume I'm mad If I would spell it otherwise.
So in short - in a formal(ized) environment, use the prescribed spelling alphabet. In everyday situations, use any alphabet you feel comfortable with, or simply use any word or name you can think of for a letter.
In my experience, people working call centers are not using any standardized phonetic alphabet, but rather their own. One is as likely to hear "B as in Brittany" or "B as in Buick" as "B as in Bob" or "B as in Bravo" Possibly some of these call centers have their center-wide alphabet. but I've heard enough strange ones to think some operators are left to their own devices.
John W. Campbell, Jr. pointed out that and alphabet of words that sound like the wrong letter might be useful militarily, because it would confuse an invading force. He published a partial alphabet such as Knave for K, Pneumonia for P, asking readers to help make the alphabet complete. I think that was about 1966, and he never published a complete alphabet before he died in 1969.