When my wife transfered from H4 to F1, SSN office said in their system, her status was still H4. I called CIS, even though CIS said you have to wait at least 4~6 weeks to change the status in the system. And after 10 more days, SSN application got through, another 10 days later, we got the SSN.
Your company will put away your salary and give it to you when you get the SSN. Don't worry.