"Best French cheeseburgers in London"

In searching (what I thought) was the video you were watching, it looks as if you can see the name of the food stand at the very beginning. Could it be called The Palate located at Camden Market? Looks delicious! Here's the link:
http://camdenmarket.com/food-drink/la-patate

Thanks, CJS! Yup, that's it!

You also might try the raclette at the Kappacasein stall in Borough Market. They use their own cheese and make a toasted cheese sandwich that I daresay would rival your gastronomic adventure with the cheeseburger. We each get one on every trip to Borough Market. Now we're going to have to follow your lead to the "best French cheeseburger in London". Happy eating.