Are you able to go to the story and download it as a pdf? (i.e. go to the story in your web browser and add .pdf to the end of the URL before the ?). If so how long does that take?
When adding .pdf to the URL before the ?, it generates a pdf in under 2 seconds. The same goes for this URL http://<SERVER NAME>/views/Test_Report/Story_1?&:format=pdf (removing the &:refresh=y).
I have solved the problem by making sure my workbook uses an extract rather than a live connection!
Thank you for your time, Jonathan!
Thanks for updating us on the solution! VizAlerts is a bit picky about performance by default; this is because it was written from the perspective of a Server Admin who didn't want Tableau Server being tasked with a ridiculous number of slow alerts running every 15 minutes! But all of the timeouts can be adjusted, across the board, or for individual vizzes, using the VizAlertsConfig workbook. So if you have a viz you're trying to work with whose performance widely varies (as is common with those based on live connections), you can work around it by simply setting the timeout higher and making VizAlerts more forgiving.