I've not flown with BA, but I've been through the new Airport.
There is a lounge. I walked in, and straight back out as there didn't appear to be much in the way of booze/food. However, I was the first person in there, and I don't think they were expecting anyone (I checked in very early for my flight).
There is a 'sports bar' next to the lounge which was OK. They were showing an old tennis match on every screen, followed by the 2001 world fencing championships

But they served cheap Kingfisher.
The airport is 40km from the city, so allow 2 hours to get there. It took me over an hour an a Sunday afternoon when the traffic was light.
There was no queue for security - but this was for an afternoon flight. During the early hours, there are more international flights.
The other thing to be aware of is that you need to pay a 1070 rupee departure tax until 7th July (when it becomes included in the airfare). You get this at a desk on the far left as you walk into the terminal. Again, then wasn't a queue when I was there, but it could be different in the early hours.
One final thing, you need a itinery to get into the airport. So if you are on an e-ticket, make sure you have a printout.
Cheers,
Rick