Hi paulthetaxman and welcome to FlyerTalk.

The British Foreign Office says that you need six months to get into Thailand, as such your problem isn't going to be the airline getting into Thailand (sorry), but rather the Thai government:"To enter Thailand, your passport must have an ‘expiry date’ at least 6 months after the date you arrive and have at least 1 blank page."
https://www.gov.uk/foreign-travel-ad...y-requirements
So it is pretty unlikely that any airline will fly your daughter to Thailand, and if they did, you may deport you on arrival.

The rules for Hong Kong are one month: "Your passport must be valid for at least one month after the date you plan to leave Hong Kong."
https://www.gov.uk/foreign-travel-ad...y-requirements

Cathay recommends this website, that seems to have access to Timatic, which is the best source of what airlines will use: https://www.iatatravelcentre.com/ - it should really be the bible of what would be allowed, though you need to insert plenty of personal information

As far as I could tell your options would be:
1. Going back to the UK via HKG, should be fine with Cathay, even if you have a stopover (HK is a lovely place to visit as an alternative to Thailand, if lots more expensive). Other destinations or routes may work too but you're going to have to check jurisdiction-by-jurisdiction
2. Wait for a new passport (potentially emergency) and then resume your travels to Thailand

Either way, a painful lesson learned but I am sorry for the inconvenience. If you had no stopover most airlines should let you get back to your home country (even if the passport is fully expired), but I wouldn't be trusting anything other than a full service carrier with a single stop -> even if you were to choose something like Qatar (QR/Qatar Airways) then you run into Doha needing the passport to be valid for 6 months, so they worry about delays.... so even if transiting, may be painful. Luckily with CX/HKG you should be ok due to the laxer rules on British passports than would be typical, likely due to history.