I would definitely try using the on-line booking tool at www.oneworld.com. If BA is the issuing airline (owing to the first segment being on BA ex-GB) I believe there may be some problem with the online tool accepting your US credit card, but have no personal experience one way or another on this.

Just a couple of notes, however. First, remember that you can't transit your country of origin at any point after leaving it. This is important with LHR because it's such an important European hub within OW. If on the way back around you wanted to go to IST or some other European destination before finishing in the UK, you'd need to avoid LHR as you can't leave the country of issue once you've returned.

Second point (and no idea if this makes any sense in terms of your work commitments) but as of now given USD exchange rates, DONE3s are about US$800 cheaper if purchased in Poland than in Britain, and around $500-$600 cheaper if purchased in Sweden. May or may not matter, but just a note. At both WAW and ARN you can use the excellent offices of AA's GSA, Tal Aviation, who are familiar with the products.

Still, I'd try to use the online booking tool in the first instance, as you can do that 24/7.