Dr. Knight is dead. You have no idea where the Panacea might be. There are more zombies than ever. You're still in Buckingham Palace - and the Kitchens await.
Climb over the barricade beyond the air vent from the previous closet. Peer inside the Kitchens and a zombie will spot you. Get back over the barricade and use it to oust the first zombie, then the second, without danger. Then wait for the three crawling zombies to come into range so you can smash their heads open. In the back storeroom you'll find a Spread Upgrade as well as a vent; go through the vent. You'll come out a door with, surprise surprise, a manhole inside! How merciful. Head through and return to the Safe House, sleep, save, and upgrade your weapons.
Prepper's still not talking to you, and if you check the manhole you'll see that you now have only two options for travel, both back in Buckingham Palace. Use the Palace Underground route to get where you need to go, and head to Dr. Knight's lab. The game unmercifully has locked one of the faster doors and wants you to take the long route.
Go through the back door, past the locked zombies. A zombie on the ground will try to eat you. Smash it in, then check the blue room. The platform across is wrecked, forcing you to go through the vents to the bottom floor. You know, the one where there were a bunch of zombies before? If you're feeling uneasy, you can snipe a zombie or two from the catwalks before descending. Either way, you have to go down through the air vents and into the warehouse area.
When you emerge from the air vents a zombie will jump out at you. Have your Cricket Bat ready and take it down normally, crazy music or not. Begin to weave your way through the maze of boxes here, taking down zombies that rush you one-by-one; they're all far enough away from each other that this is easy, and watching them crawl through shelving units is a good way to find the path to the far door. The door is blocked, but you can crawl into an air vent beside it. Do so -
- but be ready to backtrack, because the second body you see will come to life. Slide back out of the vent and smash its head, then go back in and make your way to the toilet stall on the other side of the vent. Unbar the door, then get all the way to the back of the bathroom, as an exploding zombie is in the hall. Kill it at a distance.
Go upstairs in and into the art gallery. Use the eye on the door to open it up. Inside you'll find stairs. Go down them and kill the two zombies at the bottom, then wait here for a third to come in. Check the body nearby for a Flare and the suitcase for Queen's Letters.
You're now in a series of posh corridors that have been flooded by water. Enter the water and head down the hallway. Two zombies are by the debris; crush one's head then take on the other normally. An electrical zombie will screw with your radar in here. Yaaaay.
Go into the next room and your radar will be useless. Climb on the table to get up to the air vent here. The electrical zombie is at the end, though it will vanish. There's nothing for you back in the water right now, so start unlocking this door - but stop immediately, as a zombie will drop out of the vent behind you. Kill it, then finish unlocking.
Inside is the vault. Check the briefcase outside it for Queen's Letters and behind the door for a Spread Upgrade, then go inside and grab the USB Key from the harddrive. This will spark an alarm that'll bring two zombies to you; kill them, get the key, then run out of the vault room and into the water. There are zombies waiting here for you; go past them and to the now-open door to the right of the rubble. The only one you really need to watch for is right in front of the ladder. Flee up the ladder and head through the Bunker to the Victoria Memorial path. There are lots of zombies in here, as well as a red zombie right by the exit; clear them or run past as appropriate.
More than anywhere else in ZombiU, I found the deep underground of Buckingham Palace to be rather glitchy. I died after leaving the vault, and had to backtrack through the water to get my stuff again; I feared a tough battle here, but a lot of the zombies seemed to be stuck on the walls. What's more, when I found MY zombie, it seemed to have gotten stuck on the ladder out, and there was another zombie - an exact copy of mine - stuck with it. They came to life when I got close enough, and I had so much ammo as a result that I had to leave a ton behind. Sigh. This area also seems to spawn a lot of Survivor Zombies in general, so if you need an ammo refill, this is the place to look.