Since my area of expertise mostly focuses on early Hong Kong history, I'm not too sure how this would have been handled during the late 20th century, but I have a pretty good idea of how this would be handled during the 19th century (though I admit, I'm no legal historian either).
During the beginning of the British occupation until at least WWII, the colonial government had pretty limited interactions and dealings with the Chinese community. Most historians agree that this is because Hong Kong was set up as a trading post, so the British had little interest in exploiting resources or converting the natives, etc. They were interested in gaining the most capital with the least amount of effort. Therefore, the government had little to do with the Chinese, to the extent that position of the Registrar General, who was in charge of communicating with the Chinese, as late as the 1890s was held by an officer who could not speak Chinese. The colonial government had a judicial court system that could be used to bring both Chinese and British to justice, but because it was run by the British and governed by British laws, most Chinese preferred to avoid it because they most likely had no understanding of British laws, most likely didn't know English, and the court usually ruled in favor of the British. And when the court found the Chinese guilty, the punishments were usually harsher than punishments for the British.
In these conditions, the Chinese began setting up their own local institutions to fill this power vacuum, such as the Tung Wah Hospital, the District Watch Force, the Man Mo Temple Committee, the Po Leung Kuk, and others. These organizations were headed by prominent Chinese elite of the community and would basically take care of the needs of the Chinese, from caring for the sick to organizing charities and festivals, among many other responsibilities. In many ways, they saw themselves as the voice of the Chinese and would often act as an in-between with the colonial government. In regards to disputes, these organizations would be the arbiters of justice within the Chinese community and deal with these cases in an appropriate Chinese manner, which could otherwise not be done in the British colonial court.
So we can already see that if we have a case between two non-Chinese, they would be taken to the colonial court. If it's between two Chinese, then the Chinese would deal with it amongst themselves. If we have a case where a Chinese is accused of a crime against the British or caught by the British, then they would most likely be taken to the colonial court.
Going back to your question, assuming the rape was reported, if it was committed against a European or non-Chinese woman (or man I guess?), then it would probably go to the colonial court. I don't know of any specific examples so I couldn't tell you how this would be handled. A British historian would probably know that better than I would.
However, and I'm assuming this is what you're getting at, if the rape was against a Chinese woman, and assuming it would be reported, it would most likely be taken to the Chinese led organizations (probably the Tung Wah Hospital). Most Chinese, except for the merchant elites and the upper classes, could not speak English and had no way of communicating or complaining with the colonial government. Remember, the Registrar General who was in charge of the Chinese in the colonial government could not speak Chinese either. However, the merchant elite and upper class Chinese usually knew English and many had a Western education and went abroad. Taking a case to them would be a smart move considering they knew how to navigate the colonial system and frequently dealt with the British. Moreover, because of their wealth and education, the British often regarded these Chinese elite highly and would've taken them more seriously than a common Chinese. Moreover, the Chinese elite prided themselves on being the voice of the Chinese community and would eagerly sort out the dispute as a show of authority.
I know of one example that is similar to this, though it's bound up in a lot of other issues. During the bubonic plague outbreak in 1894, the British ordered their soldiers to clean the houses and streets of the Chinese. The "sanitation workers" would move all of the belongings to the street, clean the inside of the houses, then disinfect or destroy the belongings in the streets. These workers would also inspect the inhabitants for signs of the plague. Many people were outraged by the intrusion of these soldiers, especially because they would be inspecting the neck and groins of women for buboes. These soldiers weren't medically trained so to make matters worse, they often missed the signs of plague. Considering also that they had to clean these people's houses, they probably weren't too nice about it either. To top it all off, the government gave the soldiers extra rations of cigars and alcohol to cheer them up.
The Tung Wah Hospital, on behalf of the Chinese, complained to the colonial government about the intrusion of the soldiers, specifically mentioning that women were being violated by the soldiers. Granted, the extent of abuse against women may or may not have been exaggerated considering the intense environment of the time, but this exemplifies the process of how this mistreatment was dealt with. The Chinese felt their women were being violated and the Chinese elite in the form of the Tung Wah Hospital Committee formerly complained to the colonial government and petitioned the governor at the time to stop the sanitation inspections.
However, I would assume this wasn't done very often for individual cases. Hong Kong society was very segregated and British colonialism was based on racial superiority, considering they were a minority ruling over a vast majority. British officials were highly discouraged from any relationship between Chinese women. If an official or soldier married a Chinese woman, he would lose his job within the year. Moreover, bi-racial children were looked down upon and were more or less seen as Chinese. However, there were several Chinese brothels throughout Hong Kong that were legal that British soldiers could and did frequent. Granted this isn't rape, but these factors combined, along with the general view of women at the time, probably meant that rape cases in early Hong Kong weren't very frequent.
I'm sorry I can't tell you more about modern Hong Kong. I do know that the British dominated the government even until the handover in 1997 (except during the Japanese occupation). Throughout the 20th century the Chinese-led organizations lose power as more Chinese are English educated and these organizations lose the claim that they are the sole voice for the Chinese community. I also apologize that I can't tell you about any specific cases about rape in Hong Kong, but hopefully I have enough information about the times that you have a general idea of what was going on and what maybe would have happened if a case had been brought up. Thanks for reading if you made it to the end! Sorry for the long post!
Sources:
Primary
Great Britain, Colonial Office, Hong Kong, “Correspondence Relative to the Outbreak of Bubonic Plague at Hong Kong,” 1894.
Secondary
Carroll, John. A Concise History of Hong Kong. Lanham: Rowman & Littlefield Publishers, Inc., 2007. (I super recommend this book for a general history of Hong Kong.)
Carroll, John. Edge of Empires: Chinese Elites and British Colonials in Hong Kong. Cambridge, Massachusetts: Harvard University Press, 2005.
Jones, Maurice and Arleen Platt and Jerome Platt and Michael Regan. The Whitewash Brigade: the Hong Kong Plague of 1894. London: Dix Noonan Web, 1998.
Sinn, Elizabeth. Power and Charity: A Chinese Merchant Elite in Colonial Hong Kong. Hong Kong: Hong Kong University Press, 1989.
Tsang, Steve. A Modern History of Hong Kong. New York: IB Tauris & Co Ltd, 2004.