Come on! "a" menyasszonya? The sentence is perfectly fine without that as well.
I don't understand what you mean.
"A vőlegény a templomban vár, de menyasszonya nem jön." is a perfectly valid translation of the English sentence. Yours sincerely, person #20.
Without the 'a' is more poetic/archaic. It is fine but 19 out of 20 people would say 'de a menyasszonya...'.