Considering how she is, she probably just asked politely to be queen and everyone said yes.
Princess Serenity goes from a fragile and quite possibly immature Princess in her previous life to a Queen in her current one.
I don’t imagine that she had to ask anyone to rule the planet: she used the power of the Silver Crystal to establish the old Moon Kingdom on Earth.
If memory doesn’t fail, Neo-Queen Serenity cannot transform into Sailor Moon, but her powers are comparable to those of Eternal Sailor Moon.
30th-century Earth is a reincarnation of the old Moon Kingdom and all humans essentially live like the beings from the old Moon Kingdom.