Your passports will be sent to the Bangalore visa office for visa stamping.
It's like this:
A final decision on your TRV applications will be taken only after the visa office receives your passports. Until then you have not yet been approved.
Like I have mentioned earlier, your applications will be closed/refused if you do not submit with the stipulated 30 days. And, you have not received any confirmation from the visa office allowing you to submit your passports later. So do not risk a refusal.
I'm pretty sure you will get your processed passports back before March 18 (probably by March 14-15).... if you include the letter to expedite, etc
Do submit your passports on Monday, March 6 to VFS Pune. Check for any local holidays (Maharashtra and Karnataka) before you opt for a courier delivery of the processed passports. Otherwise, you can collect the processed passports, in-person, from VFS Pune