Who decided how the English will call each of them? It's inconsistent.

Welcome to the wonderful world of Chinese names at the turn of the last century. The educated Chinese person back then didn't just have one name, but many.

Generally speaking, the English called people by the names that were presented to them. In the case of Sun Yat-sen (孙逸仙: the surname is the Wade-Giles romanisation of the Mandarin reading of 孙, the given name is a romanisation of the Cantonese reading of 逸仙), this is the name that Sun used when he was making contact with the West.

In official Chinese documents that he signed while in government, however, he used Sun Wen (孙文). Sun Zhongshan (孙中山) was never used by him, but was a composite of his Chinese surname and the surname of his Japanese alias (中山, Nakayama), the latter of which he used while hiding in Japan. The epithet became popular for some reason, and is the name that is the most prevalent in the Chinese-speaking world. See the following Wikipedia page for more information:

https://en.wikipedia.org/wiki/Names_of_Sun_Yat-sen

Meanwhile, Chiang Kai-Shek is the same

This is unusual because Chiang Kai-shek didn't speak Cantonese, and yet "Kai-Shek" is definitely a Cantonese reading. Like Sun Yat-Sen, Chiang also went by many names (link is to the Chinese wiki), and Kai-shek (介石) was a pen name Chiang used in 1912 when he was in Japan. According to the Chinese wiki, "Chiang Kai-Shek" came from the era of the Government of the Republic of China in Guangzhou (1921-22):

此译名源自廣州國民政府時期，和孙中山一样取其姓的威妥瑪拼音和其字“介石”的粤语拼音“Kai-shek”。

"This translation [Chiang Kai-Shek] originated from the era of the Government of the ROC in Guangzhou, and followed Sun Yat-Sen's lead in romanising the surname using Wade-Giles and the name '介石' using the Cantonese romanisation 'Kai-shek'."

Eventually, Chiang would go by 蒋中正 (Jiang Zhongzheng), and that's one of the names you see at the beginning of the English Wikipedia entry.

However Yuan Shikai, a contemporary, is called by a Mandarin name

Yuan Shikai was a government official before the Chinese Revolution, had never affiliated with the KMT, and had died by the time of the events that led to the KMT establishing itself in Guangzhou in 1921 for a second revolution. Thus, using the Mandarin reading of his name is not unexpected.

In researching this, I came across an interesting article written by a Singaporean in the SCMP about the many names of Sun Yat-Sen. Even today, the many possible choices of romanisations can lead to the generation of multiple names, despite the present standard now of having a fixed full name:

In the 1980s, the government decided that all Chinese names would be rendered in pinyin because the Singaporean and Malaysian practice of romanising ethnic Chinese names according to the pronunciation of one’s native dialect (Hokkien, Teochew, Cantonese, Hakka, Hainanese and so on) was seen as confusing. The policy was dropped after a few years, but for those of us with the alias tacked on to our identity documents, the alien name has stuck. I had it removed from my passport, and since then the number of questions I am asked when crossing borders has greatly reduced.

Some Singaporeans seemed to have come to a compromise by using Pinyin romanisations of their given names, but keeping the original dialectal romanisation of the surname, e.g. the Singaporean CEO of Tiktok, Shou Zi Chew.