Gdalwarp is something of a pain. I got decent results with:

gdalwarp -r lanczos -srcnodata None -dstnodata None -multi -tr 1 1 WA_Hamilton_20200305_TM_geo.tiff WA_Twin_Sisters_Mountain_20200323_TM_geo.tiff both.tiff

Note that this changed the resolution (roughly doubled it, in fact).
There does not seem to be a good, easy way to tell it to preserve
resolution.  You must play with -tr. Anything higher than 1x1 is nasty
and coarse. Anything less gobbles disk space exponentially.

Another alternative: just drop gdalwarp and for each raster layer, set
Properties... Additional no data value to 0. Import via the following
procedure:

http://www.qgistutorials.com/en/docs/raster_mosaicing_and_clipping.html
http://www.qgistutorials.com/en/docs/3/raster_mosaicing_and_clipping.html