Now that sanctions are officially lifted (as of today), the only thing blocking carriers from serving Iraq is the fact that the airports aren''t open.
According to the DOT website, UA has existing authority to serve Iraq, as do DL and AA.
BA, AF, AZ, and LH have existing rights which were used up until 1990, so should either of them resume service to Baghdad, my guess is that you''d see the DL*, AA* and UA* code on those services from Day 1.
KL may have rights, but didn''t fly into Baghdad between 1982 and the Kuwait invasion as far as I can tell. That may prove to be as much of a stumbling block as getting DOT approval.