It would, just not in the way that you think.
Remember that people move around from area to area. people also change tenure. Some renters buy a home, and some owner occupiers sell their home and rent. Maybe not you, but other people do. Everyone who moves into a home leaves another home somewhere else. And someone else moves into that home, and so on. It's obvious, right. When people move into a new home they might move into one slightly more expensive than the one moved out of. It's often called the 'property ladder'. So when large expensive houses are built at the top of the 'property ladder' it reduces the price of large expensive houses, and then all the people in slightly less large expensive houses can move up the ladder, and then the people on slightly less expensive houses than them can move up, and so on and so on. Eventually there is less demand for the very smallest cheapest housing, and hence prices will fall. All house building, no matter the tenure, helps reduce the price of all housing.
If everyone in London could agree to build more housing, then we'd all be much better off. All of us. Rich people, poor people and those in between.