So! Tickets finally booked, with thanks to all the helpful suggestions from forum members here. But it turned out to be a bit of an adventure. In case this is useful to someone else, here's what happened.
Decided to go with two (nested) RTWs, one ex-LHR and one ex-ICN. After polishing the itinerary, I decided to try using the OW RTW booking tool. Since one ticket would not be much good without the other, I built the itineraries simultaneously on two different computers, and hit the "book" button on both right around the same time.
The Korea one seemed to work pretty well. I was quoted the correct fare in Won, got a prompt to enter my CC info, and it told me I'm ticketed and ready to go. Excellent! But, turns out it was not quite so easy. Shortly after, I got an "urgent" email from an employee of Qantas Korea, saying that I needed to provide a copy of my passport and credit card. Hmmmm. Long story short, I verified through Qantas that this was indeed a legit request. Once I complied, I was ticketed in short order.
The LHR ticket was another story. Instead of a prompt to enter payment, I got a statement that I'm reserved, a PNR, the correct price quoted in pounds, and instructions to call BA for payment and ticketing. Hmm, that made me a bit nervous after reading some of the stories . . .
First call was to BA's US RTW desk. There was a holdup when they saw it was quoted in pounds, so they transferred me to a rep in the UK, who I have to say was extremely helpful and nice. However, there was a problem on her end too; their system is not set up for credit cards with a U.S. mailing address. She said she'd try to get this taken care of, I should call back the next day. I asked how, and she said I should call the US RTW desk and ask them to transfer me to the UK.
Bright and early the next morning, I did what the UK rep suggested. Um, I think that was my first rookie mistake. The US rep wouldn't transfer me to the UK, even though another rep had done exactly that. Instead she said they could take care of the payment by converting the fare to dollars. I asked, "will that be the fare I was quoted, converted to dollars?" She said, "I imagine so. Call back in an hour." Rookie mistake 2, I should have been wary of "I imagine so."
I hung up, and a couple of minutes later, what do I get but a call from the very helpful UK rep, saying she thinks she's got everything sorted out and can take my payment. I told her about my contact with the US folks and she said "let's try to take care of it and see what happens." Well, we tried several different options but her system would not accept my credit cards. She suggested I buy the ticket either at the ticketing desk in the U.S. airport, or when I arrived in London. I mentioned that the US people were trying to convert my fare to dollars, and she looked and said, "well, it looks like they've done that already." When she told me the amount, I realized with a sinking feeling that this was the price for ex-US, not the ex-UK fare converted to dollars. A not insignificant difference.
So, time to be creative. I called the US desk, told them the rates did not match, and asked them to convert it back to pounds. Then I headed out to the airport to talk to a human being face-to-face. The woman at the desk "got it"--she looked at the history and figured out what had happened and why it was an issue. She called someone at the rate desk (? or whatever they call it) and I have to say this guy was apparently a true hero. He had to jump through a number of hoops, doing conversions on each leg separately. But, a few hours and a couple of additional phone calls later, finally I'm making payment for the correct fare with a US credit card. Whew!
To give credit where it's due, out of my substantive dealings with the BA folks, five out of six were either extremely helpful or at least tried to pass me in the right direction--they were just up against a system that made it hard for them. The remaining one unfortunately sort of gummed up the works. Kudos to the Qantas folks too; they immediately understood why I was concerned, said they were concerned too, and worked to address my issue.
So, depending on your needs, booking thru the RTW tool might work well for you. But if you're buying an ex-UK ticket from elsewhere, one of the other options, such as trying to ticket via AA RTW or a travel agent, might make for smoother sailing.