Affinity Publisher will always use superscript and subscript glyphs from the font if the glyphs are available. if these are not available, you can adopt faux superscripts and subscripts instead that scale and reposition the original glyphs.
To use font-derived superscripts and subscripts:
From the Text menu, select Show Typography.
Enable one of the options in the Figure Position section.
To use faux superscripts and subscripts:
From the Character panel, choose a superscript or subscript option in the Positioning and Transform section.