Thanks to @lessonz for PR#11, included in Hudson Fitnesse plugin 1.9.
This change still doesn't handle the case where no java installation is configured at all (as described in the steps to reproduce section above).
Jenkins can handle this situation by falling back to the default java shell command.
You can, for example, perform Maven builds under theses conditions.
In such a case, the JDK selection is missing from the Execute fitnesse tests build step configuration. Submitting the configuration leads to a blank page with an unspecific exception being logged to the console, see fitnesse-stacktrace.txt.
Release 1.9, however is a big improvement as it resolves the issue for the case with one installed JDK.
So, this issue should be changed to something like "use system JDK if no JDK is configured"