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You could have your script "remember" the last line examined in a file eg /srv/nginx_monitor/last_line

LAST_FILE=/srv/nginx_monitor/last_line

END_LINE=$(wc -l < access.log)
START_LINE=0

if...

START_LINE is the last END_LINE, so it is rather
(NR>S && $6 ~ /cloudservice/ && $10+0 > 1.0) {print $10, $6}' access.log
together with the first requirement.

gawk '$6 ~ /cloudservice/ && $10+0 > 1.000 { print $10, $6 }' access.log | mailx -s "Subject" user@domain.com