We should first remember that「腺」's pronunciation derives from「泉」in Japanese (where both are pronounced sen), not Chinese. As far as I can find, Chinese sources never actually defined「腺」's pronunciation as being derived from「泉」in Chinese.
The source material from the Shen (2011) reference (First report of the committee on medical terminology appointed by the China medical missionary association: Terms in anatomy, histology, physiology, pharmacology, pharmacy) does not appear to be available on the web, but there is no explicit indication that they adopted the pronunciation 「泉」. This is what the reference actually says:
The terminology committee thought that the translation term used in Japan for
“gland,” 腺, pronounced chuan in Chinese, could denote a “flesh spring,” and
thus the committee considered it an appropriate translation term.
Or, in the Chinese reference (download link):
Gland 的译名，术语委员会认为来自日语的“腺”音 Chüuan，会意 flesh spring，是准确的
What's more, even though the committee report seems to have used the Romanisation Chüuan, the source material text smells more like an orthographic translation from Japanese.
Last part of the preface to the Japanese source text『西說醫範提綱釋義』by 宇田川 玄真 (Udegawa Genshin). The definition of「腺」is boxed in red:
Without paying attention to Japanese sounds, one might directly translate this as
「腺」 , newly coined character. Read as Chüuan.
by transcribing「泉」into Chinese sounds, because the target language is, well, Chinese. But First report of the committee on medical terminology is an English text!
You may have a look at Chinese materials which explicitly define the pronunciations:
Or more rarely,「ㄒㄨㄢ」, which seemingly sounds closer to「泉」, but this is from the Taiwanese Hokkien dictionary 《彙音寶鑑》 (rather than the national language), and in Taiwanese Hokkien「線」may also contain an「ㄨ」(Hokkien, POJ: sòaⁿ / siàn).
My explicit take on the questions:
Japanese seems to be the only language where the reading is consistent across all three characters. Why is this the case?
This is because Japanese is rather phonologically poor; the distinctions made in the other languages have merged in Japanese.
Is there a connection between 「腺」 and 「線」?
Not in terms of meaning. The original logic behind choosing「線」for the reading of「腺」may be lost; all we know is that Chinese dictionaries have defined this reading early on, and some Chinese topolects may pronounce them similarly.
Or was there simply an error when 「腺」 was introduced to China/Korea/Vietnam?
I have a few speculations but no hard evidence:
- Chinese adopted an imitation of「せん」as the pronunciation of「腺」.「線」is not that far from「泉」in several topolects, so I doubt that it would have been seen as "weird".
- An early definition like 讀若線 in an authoritative dictionary like 《中華大字典》 has large implications on later dictionaries, and may have influenced Korean and Vietnamese adoption of the term.
- For Korean:
- It may have adopted the Chinese definition (meaning & sound) directly
- Interestingly, the 訓讀 of「腺」(샘, saem, also the native word for spring as in water source) is coincidentally close to Japanese sen in initial and vowel
- Vietnamese may have adopted straight from Chinese (and please note, tuyến and tuyền only differ in tone; there is not as much discrepancy as in Mandarin)