Yeah, there is a websocket connection and when the transcription is done it will update the frontend with the links to .srt, .vtt and .txt file downloads
On Firefox 102.4.0esr, also uBlock Origin.
edit: don't know if you'll see this any time soon, but I've had it fail/hang again. You might want to take a hash of uploads, so if the lost connections still end up getting transcribed, if they're reuploaded they won't get transcribed again.
Also I haven't had success in Firefox, only Chromium.