"Reasonably" is doing a lot of work there. It sounds an awful like you're defining the boundaries of reasonability at "Exactly what is built right now and using the exact borders of current municipalities", which is a tautology. Even if we limit ourselves to the current municipal boundaries of London, population density is ~6000 people per square kilometer. Kowloon Walled City survived with 3,000,000 people per square kilometer, Manhattan has 75,000, Dhaka 23,000, San Francisco 19,000.
Their system is a much better one than we tend to have in the US, but "All-in on public transit" looks more like Trantor than London. A majority of the TFL system was constructed more than a century ago.
Nope. London is reasonably full already. The problem is that it's already 9 million people, and the high-paying jobs are focused in a fairly small area.
So if you increase the population density, the transit to that focused area will have to carry more people. And it's already at capacity.
If you want to increase the density, then London will have to create new business areas outside of Canary Wharf, City, and Westminster. At which point, the question becomes "why even bother with London?"