Yes, Nilgam provided the VFS tracking number once they handed it over, you can check the status on VFS's website.
For the time, I can say they did OK, since we tried right after first flights were allowed from Iran to Turkey.
After we gave them our passports all flights got canceled again. We had to wait for quite some time. After the flights were picked up again, they called and said they will send it out. The only problem was they didn't say when the flight will be (maybe its more my fault for not asking) but after about two weeks they gave us the tracking number. It think this two weeks could've been shorter, but given the circumstances I don't have any major complaints with them.
By the way sorry for rambling on, its been so hectic I'm having a hard time piecing all the events together.