I prefer BA in the US over the UK, because they have a dedicated RTW desk, although you may have to be insistent in order to be put through to them (some staff will insist they can't put you through). I've found BA telephone staff in the UK to be depressingly inept when it comes to RTW tickets.
As for availability, when expertflyer.com has shown D seats available with BA, AA and QF have both in recent past insisted there were none and waitlisted me, so I had to call BA and, sure enough, they could confirm my seats.