Discovered that caisd-rest.war was missing from Service Desk Manager\bopcfg\www\CATALINA_BASE_REST\webapps - it was suspected that it had been deleted by accident.
To repair it, copy caisd-rest.war.bck from Service Desk Manager\site\rest to the folder, then run
pdm_tomcat_nxd -c stop -t REST
pdm_tomcat_nxd -c start -t REST
After this, the REST application calls are working again.