You can do it like this:
start_time=$(date +"%s")
while true
do
elapsed_time=$(($(date +"%s") - $start_time))
if [[ "$elapsed_time" -gt 1200 ]]; then
break
fi
sleep 1
if [[ $(grep -c "specific string" /path/to/log/file.log) -ge 1 ]]; then
break
fi
done