Based on multiple people's advice I posted it to Bangkok via DHL. It took 2 days to arrive. For anyone else with this problem the fasted way to resolve it is definitely to get someone to send the BRP card to you.
I'm a British citizen and my wife is South Korean. We've been living in England and she works in England on a spouse visa. We went on a 2 week trip to Thailand with Finnair and on the return from Bangkok they wouldn't let her board because she didn't have the BRP card. We knew she'd forgotten it but we had no idea this would prevent her boarding, we just thought she'd have some hassle at UK immigration but once they checked her fingerprints, iris, face recognition etc she'd be fine. She can legitimately stay in the UK, we just forgot the card.
So now she's stuck in Thailand and I've had to fly home because I have work. It's weekend so the embassy isn't open and the immigration department can only be called on weekdays. Because she's not a British citizen she can't call the emergency hotline either.
The plan so far is for her to wait until Monday and then visit the UK embassy to get advice. It's possible that I will have to post the BRP card to her but we're going to wait and see what the embassy say first.
This is ridiculous because the point of a BRP card (Biometric Residence Permit) is that they've taken all her biometrics and keep them on a database. She shouldn't need the card when her fingerprints are proof she is who she is. I don't understand why this is such an issue.
Does anyone know what the best thing for us to do in this situation is?