What helped was going into File | Project structure and adding a library from Maven (via search for org.projectlombok there):
That downloaded the missing jar into the local repo which I assumed should have been done by the Lombok plugin.
Seems to be yet another IntelliJ Idea bug or at least a shortcoming not clearly mentioned in either Lombok or IntelliJ plugin documentation. I've opened an issue here, perhaps this will save some struggle to other developers in the meantime.
