What resampling method is used in the raster 'Save as...' dialog?

If the QGIS library is using GDAL for a backend, my guess (without checking QGIS source!) is that it's using nearest because that's the default for both the gdalwarp and gdal_translate utilities.

In the future, and if possible to request that the process be controlled manually and also documented with the parameters you need.

For example, if the person warping or changing resolution could use Raster > Projection > Warp (Reproject)... or Raster > Conversion > Translate (Change Format)... where both dialogs can be edited to include -tr (change target resolution) or -r (choose resampling method).

To make the changes like in the screenshot provided, click the pencil button near the bottom right of the dialog.