The Covid-19 pandemic in 2020-2021 generated almost universal shocks to the interrelated complex systems of politics, society, economy, social care and health. The objectives of this article are to analyse health systems conceptually (section two) and empirically (case studies of the UK and Russia) in order to understand: their interactions with the other four complex systems; their readiness for the Covid-19 epidemics and resilience in coping with its challenges; and the impacts on them of changes in health priorities, shortages, and rationing. Evaluations are made in section three of the readiness of the UK and Russia health systems in January-February 2020 and in section four of their resilience during the First Wave (March-June). Section five assesses developments in readiness in July-August for the Second Wave and then evaluates the resilience of the health systems during the accelerations of the epidemics in September-December. The concluding section discusses the modulations of infections and vaccination programmes in 2021.