The current top (and only) knife only run has a few instances of loading from a save which saves time since it rewinds the in game time. Is this allowed?

If they're reloading from a save for example, when they mess up, it certainly should NOT be allowed in my book. Same goes for dying, you shouldn't be able to load a save, only "Retry" because "Retry" keeps the time you had when dying, loading rewinds the clock.

Do you have time stamps of when this happens in the said video?

I noticed him save but he only had 2 retrys in total both of which are on the final boss. Must have missed that part if he did infact load a save. But yeah the rules state single segment ONLY so if he did load his run is not valid.

Saving is fine as long as the save isn't loaded.