As cities continue to expand it has become crucial to describe their evolution in time and space. Building on analogies with biological systems, we propose a minimalist reaction-diffusion model coupled with economic constraints and an adaptive transport network, describing the co-evolution of population density with the transport system. Using a unique dataset, we reconstruct the evolution of London (UK) over 180 years and show that after an initial phase of diffusion limited growth, population has become less centralised and more suburban in response to economic needs and an expanding railway network. The coevolution of the rail system with a growing urban population has generated a transport network with hierarchical characteristics which have remained relatively constant over time. These results show that urbanisation patterns largely depend on the evolution of transport systems and population-transport feedbacks should be carefully considered when planning and retrofitting urban areas.