What should I do if my passport expires in less than 6 months while on a DTV visa in Thailand?

Depends on what you want to do and your resources. Maximizing what you do, you could fly back to Thailand, get your ID and new passport (or do the latter in Jakarta) and return to Jakarta for the rest of your stay then come back to Thailand. The cost of this is a round trip flight Jakarta-Bangkok and accommodations in Bangkok.

As others are saying, there are other alternatives: return now to Thailand and settle things in Bangkok (loses out on rest of Indonesia), stay in Indonesia return on a visa exempt stamp and have to get a new DTV, negotiate a way to get a new ID sent to Jakarta (you are not certain it will work) and handle things before you return, enjoy Jakarta but return to the UK to sort things out before returning to Thailand.

None of these are ideal. You have to assess their probability of success, costs (financial and otherwise), and how they work, and decide which way forward you want to go.

I'd fly back to Thailand while you can still get stamped in and sort your passport out, if you have time then you can go back. I wouldn't want the hassle of trying to sort all your id out of thailand

Thailand is one of the few countries in the world that does not require your passport to be valid for 6 months in order to enter.

BUT the rule that Thailand goes by means you're still not allowed to enter. Because your passport must have enough time left for the stamp you will be entering on. A DTV gives 180 day stamp. So if your passport is not valid for 180 days then you're not eligible to enter Thailand. You'll either need to renew your passport in Indonesia or you'll need to return to your home country and renew it before returning to Thailand.

They don't care that you want to enter Thailand to renew your passport. That's your problem, not theirs. You're simply not eligible. Your airline won't even let you board your flight.