I use JIRA installed at a customer site so I cannot install plugins or run Java.
I want to calculate total time each issue was actually assigned to me, so I need to:
Can I do this with JQL only?
No, you need the script-runner, or another addon that can support scripting, or an addon that implements the reporting you need. However, if you're happy to scrape information, that's easy. Use JIRA's REST interface to run a query for the issues you want, then you can parse, reformat, recalculate and represent the issue data that's returned in any way you want on another machine.
Well, for Excel, you could start with the excel export of your issue list, but that won't contain much of the data I suspect you need. For grabbing stuff over REST you need a bit of "fetch data" code, but I think you'll get all the data. The obvious place to start is the docs at https://docs.atlassian.com/jira/REST/latest/ but I find them far too "reference material" and much prefer https://developer.atlassian.com/jiradev/api-reference/jira-rest-apis/jira-rest-api-tutorials/jira-rest-api-version-2-tutorial and https://developer.atlassian.com/jiradev/api-reference/jira-rest-apis/jira-rest-api-tutorials
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