Agadez to Dirkou (about 40 km north of Bilma) is a key transport route to Libya. You might be able to pick up a ride on one of the 10 wheel drive Merc trucks that ply this route. I would say it takes 2 to 3 days from Agadez to Bilma in one of these trucks. It took us 24 hours driving in an old L/Rover.
A lift from Dirkou to Bilma will probably take half a day of waiting in Dirkou to organise.
Not many tourists will have ever got to Bilma this way though. You'll be an unusual passenger on one of those trucks.
[This message has been edited by Richard Washington (edited 15 September 2005).]