OK, problem fixed for us. This was rather subtle. Some time ago we'd configured a user as the "Sender" in SPRO=>GRC=>GeneralSettings=>Workflow=>MaintCustNotificationMessages to be "BATCHBC". This ID was created in our test system a few years and versions ago, but somehow we missed creating it in production. However that did not stop the firefighter workflow notifications from working on SP08 and earlier versions. Apparently SP13 has more stringent checking, because it did not like non-existent ID. The SAP person referred me to OSS Note 1755815. It's not released so I haven't read it yet but as soon as I created the ID the emails started working again.
To me the most frustrating part was that I had looked in SWI1, GRFNMW_DBGMONITOR_WD, SLG1, ST01 and had traced and debugged and still not managed to find any indication of the problem. I cannot see why the application doesn't log some kind of error message somewhere instead of just not sending the configured notification email when the firefighter review workflow is invoked.