Whenever the teacher deems it appropriate.
But what if the teacher deems it appropriate to promote a student to 1st Dan after one year? I think the idea of similar rank standards for different arts is impossible. Teachers are individuals who have different ideas of what rank represents. Some think it is nothing, others everything. I forget the teacher's name but I seem to remember the Isshinryu founder promoting several of his American marine students to 9th and/or 10th Dan after a couple few years of training.
I view seasoning time as particularly important with shodan. The color black should be a real barrier in my opinion as lay people have no conception of dan levels. Make sure the student is up to snuff before awarding them that crucial 1st dan.
Why is 1st Dan crucial? After all it is the lowest rung on a system that goes up to 9 or 10. Why put an extreme standard on that?
As for the relation between competition and dan ranking, I don't necessarily believe there was originally a correlation, at least in karate. I know about the game of go and how Kano came up with a similar system for judo. But in Shotokan and Shito-ryu karate, when dan rankings first became implemented there was no tournament competition in kumite nor kata. This was added later.
I will say that Funakoshi Sensei respected Kano Sensei to the point where he would bow whenever walking past the Kodokan. And according to the original students of Funakoshi Sensei in Japan, sparring was something that was done almost from the start of Funakoshi Sensei's move to Japan. Tournament competition was added later because Karate was in its infancy in Japan and it took time to adapt Karate to a competitive activity. But doesn't necessarily mean that the idea wasn't always there. As for Shito-ryu, Mabuni Sensei was very interested in competition. In fact, the picture in my mind of him is wearing chest guards and other sparring gear. It was his influence through Dr. YOON Kwe Byung that influenced Taekwondo adoption of the same equipment.
Even today, Taekwondo and Karate utilize the belt ranking system to separate competitors of different levels. You don't see for example, green belts competing against black belts normally at karate or taekwondo tournaments. I'm sure you yourself competed in tournaments against similarly ranked opponents. In fact, people get mad when other schools sandbag and allow their students to compete in divisions lower than their actual rank.