The COVID-19 pandemic elicited a substantial hike in journal submissions and a global push to get medical evidence quickly through the review process. Editorial decisions and peer-assessments were made under intensified time constraints, which may have amplified social disparities in the outcomes of peer-reviewing, especially for COVID-19 related research. This study quantifies the differential impact of the pandemic on the duration of the peer-review process for women and men and for scientists at different strata of the institutional-prestige hierarchy. Using mixed-effects regression models with observations clustered at the journal level, we analysed newly available data on the submission and acceptance dates of 78,085 medical research articles published in 2019 and 2020. We found that institution-related disparities in the average time from manuscript submission to acceptance increased marginally in 2020, although half of the observed change was driven by speedy reviews of COVID-19 research. For COVID-19 papers, we found more substantial institution-related disparities in review times in favour of authors from highly-ranked institutions. Descriptive survival plots also indicated that scientists with prestigious affiliations benefitted more from fast-track peer reviewing than did colleagues from less reputed institutions. This difference was more pronounced for journals with a single-blind review procedure compared to journals with a double-blind review procedure. Gender-related changes in the duration of the peer-review process were small and inconsistent, although we observed a minor difference in the average review time of COVID-19 papers first authored by women and men.