Game Strats (by @Randomno)

Chapter 2
When getting the crystal sound projector, stay quite far left and you avoid the "red crystal" cutscene. Then get a red crystal, put out the upstairs fire, get a blue crystal (or 6 if your aim's bad), and go back downstairs.

You can also skip the "blue crystal" cutscene here by shooting at the blue face through the fire. It's slightly hard to hit but you can get pretty close since staying left doesn't apply when you're not close to the white machine (I think).

Also when "Circuit Complete" appears at the end of the minigame, you can pause and resume to save a second or two. I'm still looking for any other skips from pausing.

Chapter 3 is pretty boring so far. Haven't found anything other than getting a purple crystal by running in and out of the fire, but it's really hard to do and is probably slower because you won't have enough health to jump down after getting the green crystal.

Chapter 4 is basically "skip everything". Run past stuff, get a green crystal, run back, run some more, then just remember to have a red crystal when you reach the final room. The pausing skip doesn't seem to work on the other minigame because you can't pause.

Chapter 5. Run past the second fire (ideally don't get damaged by it), run past the robot at the end. In the next room, you can skip the cutscene by staying to the right. Go up the stairs and trigger the cutscene of the robot coming out. The next bit is interesting. You're supposed to open the door with the crystal, but the robots coming through the door also open it. The second robot comes out a bit too fast to get through the door, so wait a bit on the stairs when you go down and try to run through on the third robot. Then pick up a blue crystal and run past the next robot and do the minigame.

Chapter 6 is looking like another "skip 

Chapter 6 does seem to be skip everything, or at least nearly everything. The last section before the tunnels is hard to do without getting rid of the robot by placing the dog or the computer.

The rest of chapter 6 and chapter 7 is normal. Chapter 8 is also normal, the only thing is it's probably better to pick up a few red crystals before the third cyberman bit.

The first section of chapter 9 should only require 1 red and 1 green crystal. In the room with the purple elevators, you can just jump off the platform and go through the door. This puts you on very low health though which makes the next section risky. Considering how slow the elevators are, it's probably faster to kill yourself after reaching the checkpoint than use them though.

For the 2 cybermen in the room with the lift, it's possible to run past them but it's very likely you'll get shot (and die if you have less than half your health). Shooting a yellow crystal isn't much slower and much safer. Then get a blue crystal and finish the level (but also while avoiding the cyberman in the blue door room).

Doctor Who: Return to Earth