Just to update what I found out and reiterate what people have pretty much already said here. I got in contact with the international helpline (00 44 203 481 1736) and they basically just read the information on the website back to me. My wife will either have to go to a "neighbouring country" to get the test (in practice that's the Dominican Republic, Panama or Haiti which don't exactly neighbour the US) OR she can wait until she's been out of South Korea for 6 months. We're thinking we're just going to risk it and apply this month anyway once our proof of income requirements are ready.
You will not need a TB test if any of the following apply:
you’re a diplomat accredited to the UK
you’re a returning UK resident and have not been away for more than 2 years
you lived for at least 6 months in a country where TB screening is not required by the UK, and you’ve been away from that country for no more than 6 months
This was the part of the website which confused us because you
could read that as meaning that if you're a US citizen that has
ever lived outside the US in a TB test country for more than 6 months then you need the test. The lady I spoke to eventually confirmed that that isn't the case but she had to put me on hold for quite a while to go and find out and she didn't seem very certain about it when she spoke to me again anyway! At the end of the day I think it'll just depend on who processes my wife's application, hopefully we'll get lucky! If I've learnt anything it's to not bother contacting the helpline, it ended up costing over £20 and they seemed to know less about everything than I did!