The purpose of turning the bottle upside down is so the heated sauce can come in contact with the inside of the cap to sterilize/kill the nasties. It's generally said that if the sauce is 180F, the sauce needs to be in contact with the cap for 3 minutes. I'm not a food scientist or process authority. This information is based on my process authority's recommended processing for a similar pH sauce.
After the 3 minutes are up, cool down time is not important. You can leave the bottles upsidedown in the cases for later, or turn them back out and label them right now.
edit- just saw the UK- 180F = 82C