About two or three months should do it, that's what it took a couple of years ago anyway, at the time I had to sit for the Malaysian Air Law exam in KL and then a checkride for the license validation. Obtaining the work permit was the longest as Ramadan took place during the process and things slowed down a bit further in a part of the World where time seems to be in surplus anyway.
I would trust in your new company's HR and Ops departments for your info as finding a source in the DCA may be difficult. Good luck, several months seems like a long time for answers.