White Oak is basically two companies. White Oak Armament (WOA) is the most widely known and most people have their products. They use Wilson and Criterion* blanks for most of their barrels. They recently have used Shilen blanks and in some calibers (.204) they use Pac-Nor blanks. I think they use PacNor for their Tubb barrels too.
White Oak Precision (WOP) is their custom division and they will use any blank you send them or order through them. They will use Krieger, Bartlein, PacNor, Douglas and any one of the other match barrel makers of your choosing. These barrels take much longer to get because John Holliger turns them one at a time himself and you have the delay caused by ordering a custom blank. It's worth the wait.
*Criterion is Krieger's button rifled barrel division. As a rule Criterion barrels are only available to people in the industy (manufacturers). Krieger, however, will sell one of their cut rifled barrels direct to the general public.
I hope this answers your question.