Pronunciation of じゅ
In Japanese, the "u" vowel /ɯ/ has a centralized allophone [ɨ] (sometimes written [ɯ̈]) when occurring after /z/ and palatalised consonants /Cj/.1 This is 'halfway between' the standard Japanese vowels /i/ and /ɯ/ (and close to the English vowel /ɪ/ in e.g. hit).
Hence じゅ is commonly realized as [d͡ʑɨ] (as opposed to [d͡ʑɯᵝ]). It may be because of this (and the contrast of its shortness with the first 'stronger' うう in 柔) that some romanizations of the early 20th century transliterated the word as jitsu.
As broccoli forest notes, the Shitamachi dialect fronts [d͡ʑu͍] to [d͡ʑi], and in the Tōhoku dialect /i/ and /ɯ/ are neutralized generally.
One encounters a centralized allophone [ɯ̈] after /s/, /t/, /z/, and after the palatal consonants (Cy), for example in the word gyuunyuuo 'milk' [gjɯ̈:njɯ̈:].
• The Phonology of Japanese, Labrune (p.25)
Early romanizations of 柔術
Note that while the Hepburn romanization jujutsu was established as early as 1872,2 this was not an ubiquitous system at the time, and even less so in works printed outside of Japan. The earliest English references to jujutsu appear in the Japan Weekly Mail, where it is spelled jitsu:
Jiu jitsu (wrestling) is also taught, but not much practised by gentlemen.
This article also contains reference to a number of other Japanese disciplines, all with the same spelling: Gei jitsu, Ba jitsu, Ken jitsu, Ho jitsu, San jitsu.
While there was a mix of jutsu/jitsu spellings in the 1880's and 1890's,3 a number of popular English books on the subject were published around 1900 which used the jitsu spelling:
Looking at the google n-grams data, it appears this is around the time that this orthography began to gain widespread usage outside of Japan:
Jū-jutsu, ジウジユツ, 柔術, n. The art of wrestling, or throwing others by sleight.
Syn. yawara, jidori, sumōtori.
Wrestling, n. Sumō, jidori, yawara, jūjutsu
• Japanese-English and English-Japanese Dictionary by J. C. Hepburn (1872)
As you speculate, before the widespread adoption of Hepburn the expected form jiu was indeed occasionally used:
Chin-jiutsu-sho.......... 陳 述 書
The Chrysanthemum, Volume 1 (1881)
| Jiutsu-gawa, 76, 491.
... the Japanese wrestling game of jiu—jiutsu, which physically demonstrates that if you yield at opportune moments you may turn your adversary's force to your own end and advantage.
With the quickness that came of long training in the “Jiujiutsu” school he ran in under the weapon and knocked out his adversary's wind with his head, catching him round the waist and throwing him heavily backward.