Notes
How to compate remote LastModified time with local file time in BusyBox date
For some reason my busybox date does not recognize timezone format. So I had to over-engineer it.
rem_headers=$(curl --silent --head https://remote.web/online/file.gz)
rem_lm=$($rem_headers | grep "Date:" | cut -d: -f2,3,4 | sed 's/\sGMT\s*$//' | sed 's/^\s*//')
rem_ts=$(TZ=GMT date --date "${rem_lm}" -D "%a, %d %b %Y %T" +"%s" 2>&1)
loc_ts=$(TZ=GMT date -r "/path/to/local/file" +"%s" 2>&1)
[ "$rem_ts" -le "$loc_ts" ] && echo "same or older" || echo "newer"