Solution
This problem occurs because changing the PostgreSQL database path does NOT update the "Postgres.DatabaseStatus: ready" parameter in the PostgreSQL configuration file:
C:\ProgramData\National Instruments\Skyline\Config\PostgreSQLDatabase.jsonThis Database.Status parameter indicates whether a new PostgreSQL database instance should be created at the new path. The "ready" parameter indicates that the database instance already exists and does not need to be added there.

To get the services started again, edit the
PostgreSQLDatabase.json file by removing the "Postgres.DatabaseStatus: ready" line entirely, saving the file, then restarting the SystemLink services in the NI SystemLink Server Configuration dialog.