“Without utilizing third-party scripts” — why? Do you prefer having to write such scripts yourself?
Anyway: InDesign does not have a 'preference' or a 'preset' for the vertical position of diacritics. All that Peter's (free!) script does for your n-tilde and e-acute is retrieving the correct single character from your font, and the way it looks on your screen is the way the font is designed. It's a single glyph, and you cannot somehow “move up” a part of it with InDesign — not without converting the character to outlines, at least. The vertical position is, in this case, determined by the font designer.
An alternative is to use separate “n” and tilde character, and “e” and acute character. In most fonts, the tilde and acute are available as single codes; then, all you have to do is kern them to the left (as these are typically “spacing”), and use Baseline Shift to move it as high as your client desires.
(this work-around simply not practical enough).
I'm afraid there is no “practical” way to do this. You might try to convince your client that this is the way the font is designed.
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