This is Harry Turtledove's series about Alien invasion. Anyone concerned about spoilers should stop reading this review right now. The Books are Second Contact, Down to Earth and Aftershocks.
Proclaimed as alternative history it is straight out SF to me, with the exception that the Aliens invaded in 1942. WHich pissed them off no end. See they send this probe first, about 1600 years previously, and this showed 300 million Earthlings with the most advanced having swords and horses. Then the invasion fleet attacks and finds 3 billion Earthlings with not only aeroplanes and tanks and automatic weapons but in the process of developing jets, ballistic missiles, and nuclear weapons. AND, being 1942, every country was on a full war footing and totally prepared for combat.
Unfortunately I haven't found the books about the actual war, the subseries I am reading is set 20 years later, when the world is divided into Lizard controlled areas (basically the southern hemisphere), 3 major human powers (USA, USSR, the Deutch (Nazi) Reich, a couple of smaller powers (Japanese East Asian Co-prosperity sphere, Britain) and a scattering of independent countries (Canada, Finland... generally places too cold for the Lizard aliens to live).
Sydney and Melbourne got Nuked and that was that for Australia. The atmosphere isn't too radioactive: Probably not that clean, but I imagine the Lizard bombs are cleaner than primative human ones. Interestingly, the Lizards don't seem more than 10 or 20 years more advanced than humans in 2003, except for their spacecraft. Which is making it possible for humans to catch up.
Lots of tech stealing. There is a Jewish character who escapes from England (which is becoming more and more like the Reich, which is next door and a huge influence) to Canada. He works in a Widget Works that takes Lizard tech and adapts it to human uses. He develops a moving, talking, responding teddy bear and ends up rich. It confuses the hell out of the Lizards who see that their military tech has been taken and used to create this thing, but can't work out what for.
Mention is made that Hot is now a word that means good. The Lizards think that way and it made its way into English. Actually a lot of Lizard concepts made their way into English, fun is had with this all the way through.
Alien headquarters was in the outback but it was changed to Cairo after a missile attack loaded with Ginger.
Ginger is a very powerful drug to the Lizards. It gets the guys all high, and the girls get high and horny. As Lizards NEVER get horny except for mating season, and the guys automatically have sex when they meet a horny girl, an explosion of Ginger results in mass orgies on the streets. So too, having a female addicted to ginger means that if she ventures outdoors while high any passing male will loose all control and start mating with her. A female character in the book keeps having this problem. It's a real problem when she doing ginger in her office and is unexpectedly called to her (male)boss's office to see a (all male)delegation from Cairo.
The effect lasts for a day or so. And the presence of female pheremones means that any male within a few km is all edgy and horny and prone to starting fights with other males. (The Lizards describe it as making them act the way Big Uglies (human) males act) But it was banned even before the ships containing the females reached Earth and the sex effect was discovered. It also has the general effect of cocaine.
A large part of the books is actually about the Lizards trying to stop the Ginger smugglers. It is also concerned with the stupid, insane Nazis starting a new war to conquer Poland and get the Jews there. There is also the plotting between Molotov (head of USSR) and Mao (rebel leader in Lizard controlled China)
It is mainly a cold war, with lots of trade and negotiation.
An interesting aspect of the story is how the humans find that in many ways the Lizards are better than a lot of the humans. Everyone prefers the Lizards to the Nazis for example. At the end of the trilogy the French leave German control for Lizard control. And the Fins allow some Lizard control to keep the (Communist) Russians out. The Jews and the Poles allied themselves with the Lizards to throw the Reich and the USSR out.
I recommend this highly. Harry is rapidly becoming one of my favourite authors. The previous series I read by him was that of The American Front, Walk in Hell and Breakthroughs. Alternative histories of World War 1, where it didn't turn out as well as it did in our world...