I saw the following written on Google:
A non-Jew’s Sabbath handiwork that was not done for me—that is, Sabbath work done by a non-Jew without any request or involvement from a Jew, and not intended for him—is permitted for use by the Jew. The prohibition of “Sabbath handiwork” applies only when the non-Jew did the act especially for the Jew, or when the Jew asked him to do it.

In the present case, based on this, could one permit watching the video without waiting the amount of time needed for it to be done?

Simply speaking, such a situation is considered as having been done for you, because he does it for all the viewers/customers.

In a case of loss, is it enough just to wait the amount of time needed for it to be done? For example, a peeled potato will blacken and be ruined while waiting until Saturday night, and some wicked fellow comes along and bakes it in the oven on the Sabbath. On Saturday night, after waiting the amount of time needed for it to be done, may one eat it? If they had waited until Saturday night, the potato would have been ruined. Does that make a difference?

In cases like these, what counts as the amount of time needed for it to be done? After all, some videos can sometimes take weeks of work (production, editing, etc.).

Noam, the question is how much time was devoted to it on the Sabbath, not how long it takes to create a video. The idea is that you should not profit from Sabbath desecration.

Harel, one can argue it either way. In a rabbinic-level doubt, one rules leniently.