Tips for vassalage:

1. Take cities! 

2. Try to have as much war support at the start of the war as possible.

3. Generate as much war support (by killing units, pillaging, etc.) to end up with 100 war support at the end.

IMPORTANT: for this scenario. Set your difficulty to Hamlet. IDK if this helps, but I shaved 15 turns off my score!

1. As I stated, I thought taking London was enough, didn't know he spread in the south
2. Gaining war support actually screwed me, as I made demands that stymied a vassalage, without them I'd have done it way earlier („war points“ went into the demands and the rest was too few to allow vassalage).
3. I already stated it, there was no other unit in the top island, safe some IP, once I took the city, after that I stopped caring about a fast win, that I knew was impossible to achieve (and a turn 25-ish was already „slow“).

I don't want to set the difficulty lower, except for some challenges and will probably do to unlock the 50 kills achievement. Otherwise, I stick with the highest difficulty, I want a bit to feel a bit of resistance on the opposite side and not only press the „next turn“ button. I'll miss the badge if getting it means having less fun while playing (acheviements and challenges give something at least).

Thanks for sharing tips! And Bravo for your fast victory!

Edit:
As I was playing a bit on hamlet to rapidly grab the Dia de los muertos reward that I was missing, curiosity pushed me to try the scenario at that level and I decided to finaly give it a go. It is easier, for sure, than to play it at the Humankind level, no surprise, but the main challenge remains the same: rivers are probably my wort enemy on this map. That said, I ended up unlucky again and couldn't vassalize him, despite having taken London and Bordeaux and not having lost even one unit. I'd not have had a pretty number of turns to boast anyway (remains 51, so 24 + 1):

The good thing is that it was possible to redeem a challenge I hadn't completed:

Well, back to Humankind now!