From what I know AAI have contracts & bases presently in Jeddah, Madrid, Paris, along with an occasional New York base for Universal Airlines & Buenos Aires for Southern Winds ( though I hear SW are in diffuculties or have gone bankrupt) . Those tend to be the most longer term bases but things are changing all the time & they could pick up work anywhere in the world & at any time. Its a very nomadic existence working for these guys & its certainly not for everyone. They currently operate a mixed fleet of B-747 mainly 200 & 300, but also operate 2x B747-400 for Iberia at the Madrid base, I believe thats with Iberia's own flightcrew, not AAI, but I may be wrong. They also have a very successful & expanding Cargo fleet with bases in Europe, the Middle & Far East.
The Air Atlanta Europe operation in the UK will be fleeted this year with around 5 or 6 B-747-200/300 plus the B-767-200/300ER's & B-757-200's, which operate for associate airline Excel Airways. Its a good mixture of long & short haul flying for these guys with basis at LGW & MAN, but I believe its a different contract than AAI, certainly from the Cabin crew's point of view anyway.
I worked for them in the past & they were ok, a bit different from your usual conventional airlines, but its certainly interesting with a good variety of work & bases around the world.
To answer your question about flight deck promotion, I understand that it is quite quick with AAI/AAE. They have alot of expansion planned & due to the nomadic existence they tend to have quite a high turn over of staff.
Speaking to friends in the cabin side of AAE, the company will eventually introduce more & more B-747-400 into the passenger fleet to replace the 200/300\'s, although it is always dependant on getting the right machines at the right price. The 400 is still an expensive bit of kit.
Bases that are probably not recommended would be in my opinion Jeddah, its abit unstable around there at the moment in terms of personal safety. Also alot of the AAI guys seem to dislike being in the UK, but at the end of the day its purely a personal choice.. What works for one may not work for another, but at least you have that choice & flexibilty with these guys.
Good Luck if you decide to give them a go