Anaconda mini-game in 1990 Oblask Dam Siberia level
Play the level on the normal difficulty setting in story mode. Once you get across to the other side of the dam and go into the other building with all the shelves, look to something that resembles a Nintendo 64 game cartridge on the first shelf. Walk over and pick it up. It will be in your inventory as "Anaconda". Highlight it and press A to play the game.
Astro Lander mini-game in 2019 NeoTokyo level
Play the level on the normal difficulty setting in story mode. After following the hacker to her secret hideout, use her password to enter the TimeSplitter research area. Once inside, go into the room at the end of the hall. Inside is a laptop and a group of lockers. Two of the lockers will open. Inside one is a sub machine gun. Inside the other is the arcade game Astro Lander. It is on the floor of the locker and resembles a Nintendo 64 game cartridge. Select "Astro Lander" from inside your inventory. A small menu will appear with Astro Lander and two other areas. After playing it or turning off your Gamecube, select your temporary uplink (in story mode) and press A to play Astro Lander or any other game you may have found.