The COVID-19 pandemic has prompted countries around the world to introduce smartphone apps to support disease control efforts. Their purposes range from digital contact tracing to quarantine enforcement to vaccination passports, and their effectiveness often depends on widespread adoption. While previous work has identified factors that promote or hinder adoption, it has typically examined data collected at a single point in time or focused exclusively on digital contact tracing apps. In this work, we conduct the first representative study that examines changes in people's attitudes towards COVID-19-related smartphone apps for five different purposes over the first 1.5 years of the pandemic. In three survey rounds conducted between Summer 2020 and Summer 2021 in the United States and Germany, with approximately 1,000 participants per round and country, we investigate people's willingness to use such apps, their perceived utility, and people's attitudes towards them in different stages of the pandemic. Our results indicate that privacy is a consistent concern for participants, even in a public health crisis, and the collection of identity-related data significantly decreases acceptance of COVID-19 apps. Trust in authorities is essential to increase confidence in government-backed apps and foster citizens' willingness to contribute to crisis management. There is a need for continuous communication with app users to emphasize the benefits of health crisis apps both for individuals and society, thus counteracting decreasing willingness to use them and perceived usefulness as the pandemic evolves.