The COVID-19 pandemic raised many challenges for university staff and students, including the need to work from home, which resulted in a greater reliance on technology. We collected questionnaire data from university students (N = 894) in three European countries: Greece, Italy, and the United Kingdom. Data were collected between 7th April 2020 and 19th June 2020, representing a period covering the first lockdown and university closures in these countries and across Europe generally. We tested the hypotheses that technology-related stressors (techno-overload, work-home conflict, techno-ease, techno-reliability, techno-sociality, and pace of change) would be associated with anxiety and depressive symptoms, and that coping styles (problem-focused, emotion-focused, and avoidance) would mediate these relationships. Results showed significant positive associations between techno-overload, work-home conflict and anxiety and depressive symptoms, and significant negative associations between techno-reliability, techno-ease and anxiety and depressive symptoms. A significant negative association was found between techno-sociality and depressive symptoms but not anxiety symptoms. No evidence was found for an association between pace of change and anxiety or depressive symptoms. Multiple mediation analyses revealed significant direct effects of techno-overload, work-home conflict and techno-ease on anxiety symptoms, and of work-home conflict and techno-ease on depressive symptoms. Work-home conflict had significant indirect effects on anxiety and depressive symptoms through avoidance coping. Techno-overload and techno-ease both had significant indirect effects on anxiety symptoms through problem- and emotion-focused coping. Techno-ease also had a significant indirect effect on depressive symptoms through problem-focused coping. The findings add to the body of evidence on technostress amongst university students and provide knowledge on how technostress translates through coping strategies into anxious and depressive symptoms during the disruption caused by the outbreak of a pandemic disease.