Okay, finally solved it.
For me the problem was a damaged inode in the /opt/node-red/nodes/ folder. I had installed a new node there but the directory was no longer functional. node-red was choking on it during start up. I discovered that by running /opt/node-red/bin/node-red-pi from the command line, which reported the relevant error. Running node-red-pi was a wild guess – I tried for ages to find what the actual invocation command is when you click the “Enabled” button for Node-RED in the AEP web interface. After tracing through a heap of javascript and lighttpd.conf stuff I found a reference to /usr/bin/rcell_api but go no further.
Anyway, the inode was stuffed so I moved the enclosing folder out of the way, copied everything into a new folder of the same name and I’m back in business.
Took forever partly because all the node stuff (including npm and node-red) takes so bloody long to start.