Given that a player would like to play through the games in this chronological order, which order should the branches be played? I originally figured left to right would be natural, but was reminded that the timeline page was originally published in Japanese, in which it would be read right to left. Are there any strong arguments for one of these orders, or any particular order at all?
Majora's Mask is considered by many to be a direct sequel of Ocarina of Time, taking place very shortly after the events In Ocarina. A Link to the Past and The Wind Waker both take place an indeterminate amount of time after (perhaps thousands of years), so there's no clear connection (A Link to the Past came out before Ocarina anyway).
It ultimately doesn't matter too much, but playing Majora's Mask directly after Ocarina makes sense to me since it came out very shortly after, and takes place very shortly after.
The only games in the series which act as direct sequels are Phantom Hourglass, which picks up right after Wind Waker, and Majora's Mask which takes places shortly after Ocarina of Time. Outside of those games, though, the games are barely connected at all, so there's no reason to concern yourself with play order. Frankly, even with those games, you wouldn't miss out on much by playing them out of order.