My best advice would be to create a fake appointment using the VFS online booking system to find out when the next available appointments are.

Then delete this appointment.

https://www.vfs.firm.in/pl-apptsystem/appscheduling/

When I last looked I could say with some confidence that I would have an appointment within 10 days. Best for you to look at the available appointments first then try and schedule your application.

You could always book your appointment in advance before you apply online.

DHL Express only took 24 hours for me, your mileage may vary.