From The TimeSplitters Wiki

The Game Cartridges are mini-games dispersed throughout TS2's Story Mode. After you find one, you can play the game through your Temporal Uplink at any time by either pressing the Y button (GameCube) or Triangle (PS2).

Anaconda

Anaconda is found in Siberia on any difficulty mode. It is on a shelf in the first room of the building across the river from the start point.

Anaconda is very similar to the Snake game found on many cell phones. You start out controlling a small snake, collecting dots to earn points. Each dot you eat makes your snake grown in size, becoming longer and therefore harder to avoid. If the snake runs into itself or the edge of the screen, it's game over.

RetroRacer

RetroRacer is found in the Robot Factory level on hard mode only. It is in the final large room just before you fight the Machinist, under a ramp in the northeast corner.

The RetroRacer game is a simple 2D racing game. Its only objective is to complete five laps. It is also probably the only Temporal Uplink game that is really capable of being played by two players.

AstroLander

AstroLander is found in NeoTokyo on Normal or Hard mode. It's in the locker next to the one containing the SBP90, in the room with the computer uplink.

In AstroLander, your main objective is to land a spaceship safely on a landing pad, despite limited fuel. If you hit the ground or touch down on the landing zone with too much speed, the spaceship will be destroyed and it's game over.