With electric heat pumps substituting for fossil-fueled alternatives, the temporal variability of their power consumption becomes increasingly important to the electricity system. To easily include this variability in energy system analyses, this paper introduces the "When2Heat" dataset comprising synthetic national time series of both the heat demand and the coefficient of performance (COP) of heat pumps. It covers 16 European countries, includes the years 2008 to 2018, and features an hourly resolution. Demand profiles for space and water heating are computed by combining gas standard load profiles with spatial temperature and wind speed reanalysis data as well as population geodata. COP time series for different heat sources - air, ground, and groundwater - and different heat sinks - floor heating, radiators, and water heating - are calculated based on COP and heating curves using reanalysis temperature data. The dataset, as well as the scripts and input parameters, are publicly available under an open source license on the Open Power System Data platform.