To measure the severity of lockdowns you can refer to "Oxford COVID-19 Government Response Tracker". As a proxy for unemployment (prior to release of official monthly data) you can use unemployment proxies, such as employment expectations of private sector firms, published via European Commission business and consumer survey.
A positive relation between severity of lockdowns and unemployment rate increase is highly likely. Relation between the number of infected people and unemployment rate is not as clear, since severe lockdowns can decrease the former but increase the latter.
In this study, it is mentioned, among other data, the number of people which suffer to lose their jobs (Hospitality Industry)due to the coronavirus: Preprint Tourism: The Great Patient of Coronavirus COVID-2019
hmmm, it is a very good question but it would be interesting to see if a causal relationship between those two variables (infected vs unemployment rate) exists as this can example can easily fall into the category or fallacy that Latins (used to) call "cum hoc ergo propter hoc".
But, I certainly agree that it would be interesting to see some research done on this topic.