Why have SQL Server services disappeared from services.msc?
Can't explain why this cured my problem, but while using services.msc on the computer that wasn't showing the SQl Server services, I clicked on Action->Connect To Another Computer. I connected to the other DB server (where SQl Server services appear ok on its services.msc) and then I connected back to  he local computer (DB01). The SQL Server services now appear as expected