Not really what you're asking for, but Klay Oven does do pick-up, and in past experience they can have everything ready in 15-20 minutes.
Raj Darbar up on Lincoln does delivery downtown, at least as far as river north, but they are REALLY slow - I've waited more than 90 minutes for dinner. It can be a painful experience. I don't use them anymore.
Maybe India House on Grand?