Electrification of heat will impact demands on power systems, potentially increasing sensitivity to weather variability. We have developed a spatio-temporal methodology for assessing electricity demand in the context of weather variability. We analyse varying levels of electrification of heat in the United Kingdom and simulate local weather impacts with an ensemble of 100 weather realisations. Across the scenarios, the maximum simulated national electricity peak demand doubles compared to today. Assuming current weather pattern, the weather-induced variability in electricity peak is projected to range from 6.1 to 7.8 GW (10.2-15.2% of mean peak demand) in 2020 to 6.2-14.6 GW (9.8-22.2% of mean peak demand) in 2050. We find that future weather may exacerbate the impact of electrification of heat on peak demand. However, socio-economic uncertainty predominates weather-induced variability. Electrification of heat without reducing heating demands will result in dramatic increases in peak electricity demand as well as increased exposure to weather effects. Regions experiencing a combined increase in peak demand and weather variability will likely prove to be particularly challenging for balancing demand and supply. Switching to alternatives fuels such as hydrogen or measures to lower heating demand reduces the need for additional peak electricity capacity as well as mitigating impacts of extreme weather events.