Background: The COVID-19 pandemic has had a major effect on both older people with dementia and families caring for them. Methods: This paper presents the results of an online survey carried out among Italian and Hungarian family carers of people with dementia during the first pandemic wave (May–July 2020, n = 370). The research questions were the following: (1) How has the pandemic changed the lives of family carers? (2) How did government restriction measures change the availability of care-related help? (3) What other changes did families experience? Results: Results show that about one-quarter of both subsamples experienced a deterioration in their financial status. A decline in both general and mental health was also reported. Due to “lockdown”, family carers’ burden increased substantially. Utilization of care-related help decreased, and the share of those left with no help increased in both countries. Cross-country differences emerged in terms of dementia care system, severity of the first pandemic wave, and measures put in place by governments. Findings outline the weaknesses of support structures and their country-specific vulnerabilities to a worldwide pandemic. Conclusions: To better protect people with dementia in the future, it is essential to strengthen their family carers, and support structures need to be re-evaluated and re-designed.