For this one I decided to take DAZ's challenge:
1) No Kills except Target who must be killed with the Hidden Blade
2) No Detects except upon Assassination Moment
3) No HUD
However, I modified the rules as follows:
1) Knockouts of Templars is allowed but not Royal Guards
2) No Awareness Meters
3) No Smoke Bombs
Skills:
4) Evie may use the Chameleon skill
Basically I used two knockouts instead of smoke bombs. If it looks easy, let me assure you that it was not. It took about 100 takes over three days to get everything right, in large part because the timing of the guard patrols are randomly phase-shifted. More about that later.
Approach
There are three well-known entrances to London Tower: the front door, the back door, and the rooftop access to the southwest tower. I used a lesser known fourth entrance: the rooftop access to the northeast tower. It is less guarded than the other tower but has a locked door. I stole the key in an earlier playthrough. Also, the spiral staircase here goes all the way down to the ground floor (the southwest spiral only goes down to the second floor, where Lucy waits to be assassinated). If there are other entrances besides these four, please mention them in the comments. There might be an open window or something that I missed.
So I used the rope launcher twice to reach the northeast tower where I quickly switched into Stealth Mode. Before shimmying down the tower, I checked the location of the Royal Guard on the right. If he's coming around the corner, he can become aware of Evie as she drops through his line of sight.
When hanging on the edge of the roof, it's important to be behind a solid section. They can see Evie if she's hanging from an open rail. To get lined up correctly, I climb straight up from the window below.
Sneaking past the Royal Guards
There are only three Royal Guards you need to watch: the two who line up directly in front of Evie who I will call "Left" and "Right", and the one farther to your left who patrols the eastern edge of the roof.
Okay, here's where phases first come into play. Left and Right have symmetric L-shaped patrol routes of equal length. If they both arrive in front of Evie at the same time, they are "in phase" (or zero degrees out of phase). If they arrive individually at equal intervals, they are 180 degrees out of phase. Or they could be out of phase anywhere between 0 and 360 degrees. It's random. However, they will retain exactly the same phase shift because their patrols take the same amount of time. Waiting will not affect it.
Worst case: 180 degrees out of phase. It's still possible to sneak past them but you have to be fast. Wait until Left is arriving and has his shoulders perpendicular to Evie, right in front of her moving from left to right. Quickly jump over the rail and cozy up to the door. If you do it correctly, Evie will enter Chameleon mode while outside of his peripheral vision. While in Chameleon mode you can unlock the door. Evie will remain in Chameleon mode and the guards never come close enough to detect her.
Besides Left and Right, you need to watch out for the third guy at the far left. If he's walking toward you, wait for another cycle.
Best case: close to 0 degrees out of phase. When Left and Right leave just hop over the fence, unlock the door, and walk in.
Usual case: somewhere in between. As shown in the video, it usually takes two cycles to enter the tower. One to unlock the door and another to walk through it. Just hang out in Chameleon mode until the time is right. Entering the door breaks Chameleon mode and any one of the three guards could potentially detect Evie.
The Hallway of Doom
This is by far the hardest part. Separating the spiral staircase from Lucy Thorne is a long hallway with three patrolling Templars. Again we have Left, Right, and The Third Man. Left and Right have equal length patrols and thus remain phase locked. The Third Man has a longer route which actually takes him out of the hallway half of the time.
Worst case: Left and Right are in phase or 180 degrees out of phase. Yes, both are bad. If they're shoulder to shoulder, you can't kidnap/knockout one without alerting the other. If they're at opposite ends of the hallway, then when one turns his back the other will turn to face you. In cases like these I go into Chameleon mode and throw a knife into the trunk at the far end of the hallway. Sometimes this causes one guard to jump down to the first floor (score!) or at least shift their phase. I haven't found this method to be reliable.
Best case: about 90 or 270 degrees out of phase. It's perfect when one Templar leaves and then five seconds later the other one turns his back. This is a good opportunity to kidnap the one closest to you. Be sure to watch out for the Third Man.
Every other cycle, Left will stop to look out the window. This may look like a good chance to sneak up on him but it isn't. He's always looking around and can spot Evie every time. It also puts him out of range when he turns his back. Wait until he comes closer before making your move.
The Coup de Grace
After secretly knocking out Left and Right, all that remains are The Third Man and another Templar who patrols the far side of Lucy's position. You can see him from time to time looking out the window across from the far end of the hall. The best time to approach Lucy is when The Third Man is leaving the hallway and the other Templar is walking toward the window. In the video I took a chance because he just finished looking out the window and was heading back. I was lucky that some columns obstructed his view.
Remember to rotate the camera downward toward Lucy Thorne. In Syndicate it's important to highlight your target in red (by looking at him/her) before attempting an air assassination. I'm more familiar with earlier titles where you could do blind air assassinations, especially on prime targets.
Escape
This is basically the same escape that DAZ used in his non-lethal run. It doesn't make sense to remain anonymous here because all hell breaks loose as Templars fight Royal Guards and/or Assassins dressed as Royal Guards. I'm not really sure who is fighting who so I just boogie back to the boat.