So does Java provide an API for continuations or just the virtual threads hidden behind the threading API? You can't use threads to wait for something on the UI thread (which is why e.g. SwingWorker exists), but you can with await.
There is no public-facing continuations API afaik.
Structure concurrency/virtual threads seem like a good fit for Swing; just have your event handler fire off virtual thread[s] and do your work and call SwingUtilities.invokeLater to schedule the result to be applied on the event loop thread when you're done. Structured concurrency simplifies managing groups of concurrent tasks, cancelation, etc.