The Subway Chase
Throughout this chase, I recommend just firing from the hip. You won't have the time or stability to aim well, so just get them in your general sight and fire away. Go ahead and fire through the back window when you're behind it, but note that the real firefights will come when you pull alongside it. Watch the doors as you chase after it, until you take the train out in a bit of an unplanned way.
Move forward with Wallcroft and go through the wrecked train to reach Westminster Station. Take out the first few guards, and then use the map board for cover to fire on the rest. Use a flashbang to open things up, and then move forward. Use the vending machines and map boards for cover, and you'll have no trouble clearing a path.
Turn to the right and wait for Wallcroft to take down the terrorist that runs around the corner. If you're low on ammo, go ahead and grab his red dot P90. It's a good gun, and there's a lot of ammo lying around for it.
Either take cover on the corner and carefully pick off the guards from left to right, or charge forward and get behind some cover on the left wall (by the surviving civilian). Gun down the men taking cover by the vending machine, and then turn the corner.
As you approach the escalator, run to the bottom and then go prone. Aim up to the top to gun down the man above you. Wallcroft should take care of the other. Toss back the grenade that they throw down, and continue to give chase through the landing.
The second set of escalators will have more terrorists at the top. The left side should be clear though, so just run up and flank the rest. Fire at the men jumping the turnstile, then jump it yourself. The set ahead will surrender to the reinforcements, so just walk past them and take up your spot in the blockade. Fire at the window of the truck to bring it down and set up the next series of events.
You'll get a brief transfer to someone's vacation video. Wait for the little girl to approach the truck, and then walk forward with the camera to trigger the end of the mission. Note that yes, this was the disturbing content. If you turned it off earlier, then you won't see it.