From the late Ming encyclopedia Tiangong Kaiwu (天工開物)

諸葛弩，其上刻直槽，相承函十矢，其翼取最柔木為之。另安機木，隨手扳弦而上。機巧雖工，然其力棉甚，所及二十餘步而已。此民家妨竊具，非軍國器。

Now bear with my terrible translation (feel free to correct it):

"Zhuge Crossbow, it has a straight magazine on top containing over 10 bolts, its wings (the bow part?) is made up of the softest wood. With a separate wooden mechanism, it quickly loads and looses. Though intricate, its power is weak, so its range merely 20 paces. This is used by civilian against bandits, not a weapon of the armed forces."

Likewise from Wubeizhi 武備志, an actual military treatise from the same period:

東南人喜用之，然力輕而不能傷人。
此弩懦夫閨婦皆可執以環守其城，一弩連發十矢，鐵簇塗以射虎毒藥，發矢一中人馬見血立斃，便捷輕巧即付騎兵亦可執以衝突，但矢力輕必藉藥耳。

"The people from the southeast likes using it, but its power is weak and can not wound people.
This crossbow can be used by the weak and females to protect a city. One repeating crossbow can shoot continuously 10 bolts, and the head can have poison for killing tigers applied to it, and any shot that hits and sees blood man or horse will die. It is versatile that it can even be used if met with cavalry. But the force of the bolt is not strong, and it must rely on the poison."

In other words, due to weak power and low range, the Chinese themselves don't use them much by the late Ming, if ever.

Firstly repeating crossbows existed in Europe.

Secondly, "repeating" crossbows never achieved widespread use because they simply don't function terribly well. The Chinese repeating crossbow essentially consists of a magazine that allows a new bolt to fall into place every time the string is brought back, with a lever to aid drawing the string. The problem is this means the weapon needs to be held vertically to allow the bolt to fall into place, so one arm has be used to move the lever while upright. This limits the draw strength of the crossbow massively. Crossbows were frequently drawn with both hands and a foot like this to ensure sufficient strength, which obviously give vastly more power than one handed, lever or no. Also, simply attaching a magazine doesn't shorten reload times all that much. Replacing the bolt is not the part of reloading a crossbow that takes time, it is the drawing back of the string. It is for this reason that they weren't terribly popular anywhere in the world, including China.

The Chinese repeating crossbow, popularly known as a Zhugenu after Zhugeliang, who supposedly invented them despite archaeological evidence to the contrary, was a weak weapon. The early handheld design of the crossbow necessitated en mass utilization to optimize effectiveness due to its low penetration rate and shooting distance. You could say it was a weapon made especially for peasants. When grouped together it's powerful enough to be a deterrent for most human targets, especially squished together, but it was not a weapon with enough draw power to go hunting with, as was the tradition in Europe. The penetration was so low that it became common practice during warfare to tip the arrowhead in fresh dung or more rarely poison, since wounds were light. Infections and lack of hygiene for open wounds would have been the main source of casualties caused by this weapon, rather than hitting vital organs.

https://www.youtube.com/watch?v=ZDeAxTxzorE

Even when repeating crossbows appeared, larger single quarrel crossbows never went out of use. That is to say, repeating crossbows were not a direct and straightforward evolution of the crossobow, and did not negate its brethren's effectiveness due to a lack of penetrating power, accuracy, and firing distance. Even during the Qin Dynasty we have evidence of large crossbows that required being pulled from a sitting position:

http://s1109.photobucket.com/user/HackneyedScribe/media/QinStateterracottasittingfigureloadingcrossbowLintongQinShihuangpit.jpg.html

As well as large single use mounted crossbows (Song Dynasty) that required being cranked by two guys:

http://big5.hwjyw.com/zyzx/jxsc/zgls/200912/W020091203590478609139.jpg

http://elib.lib.tsinghua.edu.cn/techlibrary/image/imageware/more/military/21.jpg

https://the0other0guy.files.wordpress.com/2012/03/crossbow8cg.jpg

http://www.xenophon-mil.org/china/militarymuseum/milhist542s.jpg

http://i.imgur.com/a2UkFTL.jpg

Song Dynasty Cavalry that has both regular crossbows and regular bows:

http://i.imgur.com/LrE9ODu.jpg

So the answer isn't really as straightforward as why Europeans didn't adopt a repeating crossbow but how they would have. The design of the Chinese crossbow would have had to be introduced to them first, and then ease of design as well as cost effectiveness would have to be considered, and production would have to be taught to engineers. If repeating crossbows were that great then Mongols and other people would have adopted them. So my guess is that they weren't.