Thank you so much for posting the photo update. It already looks so much better with the old tiles taken out. I'm so sorry but you will have to take the floor tiles out in order to waterproof the area properly. Due to the age of the tiles and floor, we can't be certain if it will hold up as time moves forward. If we remove the old tiles we can then lay down a fresh coat of waterproofing and be confident in the knowledge that everything is done correctly.
Your bathroom is coming along really well, it looks great.
Unfortunately as @EricL has said, in order to properly waterproof the room the tiles will need to come up, as without installing a waterproofing membrane to both the floor and walls, the join will remain a potential ingress point.
The only way to avoid ripping up the tiles would be to screed a thin layer over the top, and then apply waterproofing and retile on top of the screed. The prep involved in achieiving this, however, is probably just as time consuming and likely more expensive than ripping the tiles up.
Keep us updated as you continue on with your porject, and best of luck!