Saying "most everyone" is much more popular in books than "mostly everyone".
To compute the distance between two coordinates most everyone/mostly everyone uses the Spherical Law of Cosines equation.
Which expression should I use?
Answer
You've left out the most important alternative: almost. Here's an expanded version of your Google Ngram: In most everyone, most is a contracted version of almost, an adverb modifying the every component of everyone. (Yes, I know everyone is written as one word, but syntactically it's apprehended as a 'pronominal' version of every. Any(one) works the same way.)
Most everyone and mostly everyone are colloquial variants; I advise you to avoid them in formal registers.
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