This doesn't sound quite right, what's one of the names?
The London Dungeon is near the Eye and has been around since the 70s, albeit in different forms and location. They moved closer to the Eye due to rail station construction in 2013, I believe? Down the street from the old London Dungeons location is the London Bridge Experience, which generally gets better reviews and doesn't appear to be owned by Merlin Entertainment. I'd avoid both, but neither appear to match your description.
The London Dungeon is near the Eye and has been around since the 70s, albeit in different forms and location. They moved closer to the Eye due to rail station construction in 2013, I believe? Down the street from the old London Dungeons location is the London Bridge Experience, which generally gets better reviews and doesn't appear to be owned by Merlin Entertainment. I'd avoid both, but neither appear to match your description.