Background: The UK COVID-19 vaccination programme has prioritised vaccination of those at the highest risk of COVID-19 mortality and hospitalisation. The programme was rolled out in Scotland during winter 2020–21, when SARS-CoV-2 infection rates were at their highest since the pandemic started, despite social distancing measures being in place. We aimed to estimate the frequency of COVID-19 hospitalisation or death in people who received at least one vaccine dose and characterise these individuals. Methods: We conducted a prospective cohort study using the Early Pandemic Evaluation and Enhanced Surveillance of COVID-19 (EAVE II) national surveillance platform, which contained linked vaccination, primary care, RT-PCR testing, hospitalisation, and mortality records for 5·4 million people (around 99% of the population) in Scotland. Individuals were followed up from receiving their first dose of the BNT162b2 (Pfizer–BioNTech) or ChAdOx1 nCoV-19 (Oxford–AstraZeneca) COVID-19 vaccines until admission to hospital for COVID-19, death, or the end of the study period on April 18, 2021. We used a time-dependent Poisson regression model to estimate rate ratios (RRs) for demographic and clinical factors associated with COVID-19 hospitalisation or death 14 days or more after the first vaccine dose, stratified by vaccine type. Findings: Between Dec 8, 2020, and April 18, 2021, 2 572 008 individuals received their first dose of vaccine—841 090 (32·7%) received BNT162b2 and 1 730 918 (67·3%) received ChAdOx1. 1196 (<0·1%) individuals were admitted to hospital or died due to COVID-19 illness (883 hospitalised, of whom 228 died, and 313 who died due to COVID-19 without hospitalisation) 14 days or more after their first vaccine dose. These severe COVID-19 outcomes were associated with older age (≥80 years vs 18–64 years adjusted RR 4·75, 95% CI 3·85–5·87), comorbidities (five or more risk groups vs less than five risk groups 4·24, 3·34–5·39), hospitalisation in the previous 4 weeks (3·00, 2·47–3·65), high-risk occupations (ten or more previous COVID-19 tests vs less than ten previous COVID-19 tests 2·14, 1·62–2·81), care home residence (1·63, 1·32–2·02), socioeconomic deprivation (most deprived quintile vs least deprived quintile 1·57, 1·30–1·90), being male (1·27, 1·13–1·43), and being an ex-smoker (ex-smoker vs non-smoker 1·18, 1·01–1·38). A history of COVID-19 before vaccination was protective (0·40, 0·29–0·54). Interpretation: COVID-19 hospitalisations and deaths were uncommon 14 days or more after the first vaccine dose in this national analysis in the context of a high background incidence of SARS-CoV-2 infection and with extensive social distancing measures in place. Sociodemographic and clinical features known to increase the risk of severe disease in unvaccinated populations were also associated with severe outcomes in people receiving their first dose of vaccine and could help inform case management and future vaccine policy formulation. Funding: UK Research and Innovation (Medical Research Council), Research and Innovation Industrial Strategy Challenge Fund, Scottish Government, and Health Data Research UK.