What happens to the recording afterwards?

The broadcast is on your own YouTube channel, so the recording is yours. YouTube keeps it automatically as an unlisted video on your channel after the stream ends, and you download it from YouTube Studio whenever you like — go to your Content list, find the video, and use the download option in its menu.

Because it lives on your channel and not ours, nobody else can take it down, and it is not affected by anything that happens to another couple. Keep the watch link saved in Setnayan and your celebration page can keep showing it afterwards.