Flooding in urban areas during heavy rainfall, often characterized by short duration and high-intensity events, is known as "surface water flooding." Analyzing surface water flood risk is complex as it requires understanding of biophysical and human factors, such as the localized scale and nature of heavy precipitation events, characteristics of the urban area affected (including detailed topography and drainage networks), and the spatial distribution of economic and social vulnerability. Climate change is recognized as having the potential to enhance the intensity and frequency of heavy rainfall events. This study develops a methodology to link high spatial resolution probabilistic projections of hourly precipitation with detailed surface water flood depth maps and characterization of urban vulnerability to estimate surface water flood risk. It incorporates probabilistic information on the range of uncertainties in future precipitation in a changing climate. The method is applied to a case study of Greater London and highlights that both the frequency and spatial extent of surface water flood events are set to increase under future climate change. The expected annual damage from surface water flooding is estimated to be to be £171 million, £343 million, and £390 million/year under the baseline, 2030 high, and 2050 high climate change scenarios, respectively.