Your primary issue is that
embedded YouTube videos are embedded, and each video is isolated in its own <IFRAME> container, which makes it difficult to try to interact with, by design (the page that embeds the video isn't supposed to be able to modify the YouTube embed).
If you're using Tampermonkey, first of all you're going to have to make sure that your userscript is running in both the host page
and the iframes (make sure that the userscript Settings General "Run only in top frame" isn't turned on, and make sure the userscript matches both URLs). Then you're going to have to have a way for the host page to communicate with the iframe, which can be achieved through
iframe.contentWindow.postMessage(data, '*') and an accompanying
window.addEventListener('message', event => { /* do something with event.data */ }) inside the iframe.
You'll need to set up bi-directional communication though, so that the iframe can let the host page know when it's done playing, and the host page can tell the next iframe to begin. So inside the iframe you'll do
window.parent.postMessage(...) and you'll need a similar
window.addEventListener('message', ...) in the host window to process the messages coming from the iframe.
Then you'll need to figure out some way of keeping track of which iframe is which, which I guess you could just do by indexing them (0, 1, 2, 3, etc) and sending each iframe a message telling it "you are [number]." Then you'd need another type of message that an iframe can send the host to tell it "iframe [number] just finished playing," and finally a message that the host can send to the next iframe to tell it "start playing."
Your
data can be any type of object, as it'll basically be invisibly run through JSON.stringify/JSON.parse as it's passed from one window to another. E.g.
JavaScript:
data = {type: 'assign_child_index', index: 0}
data = {type: 'finished', index: 0}
data = {type: 'play'}
Then your message event handlers can just check data.type and proceed accordingly.
