I got it working in two steps for the first band:
gdal_translate -a_ullr -240500 240500 239500 -239500 -a_srs "+proj=aeqd +lat_0=59.458925 +lon_0=10.564472 +R=6371000 +x_0=0 +y_0=0 +units=m +no_defs" -a_nodata 9.96921e+036 -b 1 HDF5:"precip_aeqd-1000.201408.nc"://equivalent_reflectivity_factor precip-aeqd-1.tif
gdalwarp -overwrite -t_srs EPSG:25833 -tr 1000 1000 precip-aeqd.tif precip-utm-1.tif
This does not use the geoloc information in subdataset 2 and 3, but the corner coordinates from those subdatasets, converted to aeqd with
cs2cs +init=epsg:4326 +to +proj=aeqd +lat_0=59.458925 +lon_0=10.564472 +R=6371000 +x_0=0 +y_0=0 +units=m +no_defs corners.txt >aeqd.txt
and added 500m offset from mid cell to the edge. I omitted datum=WGS84
from the projection because this conflicts with the given radius.
This is my result, with the corner points from the subdatasets 2 and 3:

For all 5545 bands, you better use a vrt file:
gdal_translate -of VRT -a_ullr -240500 240500 239500 -239500 -a_srs "+proj=aeqd +lat_0=59.458925 +lon_0=10.564472 +R=6371000 +x_0=0 +y_0=0 +units=m +no_defs" -a_nodata 9.96921e+036 HDF5:"precip_aeqd-1000.201408.nc"://equivalent_reflectivity_factor precip-aeqd.vrt
gdalwarp -of GTiff -co COMPRESS=LZW -t_srs EPSG:25833 -tr 1000 1000 precip-aeqd.vrt precip-utm.tif
but the resulting tif is 2.4GB even with compression.
NETCDF:"path/to/file/filename":subdatasetname
.