During their fight on Geonosis, Count Dooku cuts of Anakin's right arm leaving him pretty defenseless but rather than finish him off on the spot, he uses the force to push him across the room. Why would he do this? Was he under orders from Palpatine or is this just a simple oversight as Anakin can't die in the prequels?
TL;DR : There was no immediate reason to.
Count Dooku may be operating as a Sith Lord during The Clone Wars, but in reality he is never displayed as totally evil. Unlike a true Sith Lord such as Palpatine, Darth Maul, or any of the others portrayed in Legends EU, Dooku never appears to strike out of malice. All of his attacks, each of his strategies & battle tactics, every aspect of the man seen on-screen, are practical & efficient.
During the battle with Kenobi and Skywalker, Dooku easily dispatches both Jedi with little effort. Both men are removed from the battle without killing, and Dooku does not pursue the matter for a simple reason - they are no longer a threat to him. As it is, Dooku only turned to face them since they were delaying his escape.
Clarification: I'm not saying he isn't evil, just that we never see him killing out of anger, rage, or enjoyment like other Sith would. Even when he needs to make an example of someone, it's not out of malice or anger - it's simply an effective tactic for discouraging others. He kills only as much as he needs to, nothing more.
From what we see in Attack of the Clones Dooku was a highly skilled and accomplished warrior. Notice how easily he dispatched both Obi-Wan and Anakin, and how he was able to hold his own against Yoda. Then, in Revenge, even though years have passed, he still manages to take out Obi-Wan in short order. It's not until Anakin is full of rage that he's finally overpowered. He also trained General Grievous, who summarily dispatched legions of Jedi. So basically, Dooku is one of the best Sith Lords, rivalling the most powerful Jedi.
Armed with this knowledge, let's approach the Clones fight. He takes out Anakin without breaking a sweat. He can totally kill the kid, but he instead leaves him maimed and humiliated. The robotic arm is a permanent reminder that Anakin is no match for him. And, as we saw in Revenge, he wasn't... at least, until he was overcome with rage and had the Dark Side coursing through him. When Palpatine orders Anakin to slay Dooku, you can see from the look in his eye (and I remember reading it in the Revenge of the Sith novel) that he finally realizes that Anakin is intended to be his replacement.
So basically, Dooku let Anakin live for the same reason he let Obi-Wan live: to humiliate them. He knew he could dispatch them at the moment of his choosing, and they knew that as well. It wasn't until he got his own hands chopped off that he realized what was really going on.