After having analysed Watanabe and Takahashi's translations we shall proceed a decade forward (or so), and tackle those of Nagura Jirou (1907) and Kimura Shoshuu (1908), whose versions differ quite significantly and continue in the vein of Takahashi when it comes to applying distinct nuances of personality to the cast and Andersen's trademark polite wit in the narration.
The Emperor (or, rather, the demoted King):
Most tellingly Nagura and Kimura both render the Emperor's title into ousama (王様, "king", lit. "Lord King"), although Nagura at one point has the King pondering if he is worth being a teiou (帝王, "sovereign, emperor") - albeit not the standard word for "emperor."
His Majesty's manner of speaking varies depending on the state of mind that the monarch is in in Nagura's translation, going from watashi (私), to the far more regal chin (朕) and yo (余), whereof the former was and still is heavily associated with Chinese and Japanese emperors - treated as the official majestic plural. Kimura merely has him use the masculine, informal and assertive ore (己), albeit written with a far more archaic character meaning "self," onore. Both nevertheless use the speech patterns of an elderly monarch, i.e. notably using the copula ja (ぢゃ), the existential verb oru (おる・居る) and the emphatic particle wai (わい) - though these three example also fits the generic elderly sociolect that you see in popular culture, key-features themselves of the Hiroshima dialect.
Nagura's translation takes the form of a bilingual edition where the Japanese text faces an English one that Nagura then proceeds to gloss and offers his own translation. Kimura however is straight up a standard translation in that regard.
Nagura's translation had me initially fearing that not all of the story would have been translated owing to his fragmenting approach towards it, but my fears were assuaged when I discovered that this was not the case. His approach is nevertheless far closer to the English translation offered at hand, which in turn takes one or two liberties with H. C. Andersen's original source text, but is overall itself faithful.
Nagura has the King - in this version - use three pronouns on different occasions:
"Watashi wa kaimu mienai. Korya tadagoto de nai. Jibun wa baka nano kashira. (Sore tomo) teiou ni tekishinai no ka. Sou atte wa wagami ni totte futatsu to nai daikyoukou."
"I can't see bugger all! This isn't something trivial. I wonder if I'm an idiot? Or, am I unfit for the office of sovereign? This is the most scary thing that has befallen moi!"
Note that I use moi as a translation of wagami (我身, "myself personally") for the sake of nuance and flair.
Here the King, in an inner self-monologue, exposes his fears and refers to himself with watashi, a mature if aloof first-person pronoun that is nevertheless far more humble than his later regal self-references.
He also uses kashira (か知ら, "I do wonder"), which at the time and still in some dialects is gender-neutral.
"Aa, migoto na mono, chin no manzoku kore ni suginai"
"Ah, a magnificent thing! This, by far, surpasses Our expectations!"
Here appears chin (朕, "We, the monarch/emperor"), famously used by Chinese and Japanese emperors of old - itself an Old Chinese pronoun originally used by most, but later on chiefly by emperors. This mirrors the English translation offered above, "It has Our most gracious approval," thus prompting Nagura to translate this with chin as the mode of self-reference, over His Majesty's usual watashi, here communicating that he is trying to regain his usual royal self-composure.
"Yo ga fukusou ikani mo yoku niau, de wa nai ka"
"Do these garments verily not fit well upon one as I, no?"
The more lordly yo appears two times, in the text, both in instances where the King is pleased with the delusions of the clothes. Whilst not as imperial as chin, it nevertheless is an oratory pronoun used by real-life nobility and in fiction has been used in the translation of Emperor Palpatine's dialogue.
Thus like a peaking diagram, the King starts with regular watashi, reaches the sovereign climax with chin and then slumps to the stately if less imposing yo.
His watashi could alternatively, due to the lack of furigana, be read as washi thus truly turning him into an elderly monarch becoming bothered, shaken in his royal self-confidence in regards to these Swindler's and their fantastical cloth.
His mode style is however unflinchingly according to formalia:
heika (陛下, "Your Majesty")
The usual style for monarchs (kings/queens and emperors/empresses), literally meaning "below the throne," and referring in ancient times to the in-between courtier who would transmit messages to the Throne personally since commoners had little right to face the exalted presence of a monarch. Similarly, in the Takahashi version he was referred to with denka (殿下, "Your Highness", lit. "below the palace [halls]"), referring to the courtier this time outside of the palace rather than the throne - presumably sentiments were still that heika as a style befitted actual real-life monarchs? I cannot know, unfortunately, at this moment. Nevertheless, the last in the line of styles - for the sake of clear explanation - is kakka (閣下, "Your Excellency" or "Your Lordship/Ladyship", lit. "bellow the palace/tower") moving as far as possible away from the regal presence and giving the furthest courtier the duty of relaying messages.
Kimura tackles His Majesty with:
"Ore no me ni wa nani mo mieyashinai, shite miru to ore wa baka mo shirenai, kore wa nani ni shite komatta koto ni natta" to omoi deshitaga, (...)
"My eyes can't simply not see a thing, at all, then would that mean that I'm a fool, these are most troubling words" were his august thoughts, however (...)
Speaking like a regular bloke if with the archaic onore as the kanji for ore, which historically was the etymological predecessor for the assertive male pronoun, originally a self-deprecating and humble first-person singular pronoun and also a less-so second person singular.
The narrator uses o'omoi, the honorific form of omoi, itself as clipping of o'omoininaru (the gentleman/lady thinks), this is the tone throughout the story in terms of the narrator using a courteous language and applying the right keigo when referring to the King.
The Old Minister:
The Old Minister is in Nagura's version called Saishou (宰相, "Prime Minister"), and is typically used for "Prime Minister" or in historical contexts for the personal adviser of the Emperor of China. As such he is named as the Emperor's foremost confidante and trusted courtier. He uses watashi, the gender-neutral pronoun to convey a formal personality - though he also uses washi which is the generic oldster pronoun:
"Moyou to ii, iroai to ii, kono orimono ni washi ga mattaku kanshin shita shidai wo kanarazu heika he gonjou ni oyoba zo."
"These patterns, these colours, this fabric is indeed most pleasing and I shall most certainly and accordingly convey this to His Majesty, I dare say!"
His language is overall very masculine with zo, an assertive masculine emphatic, hence "I dare say" in the translation.
Otherwise he does not diverge all too much from the other translations.
Kimura's version of the same line goes:
"Oo korehodo mo kirei ni oreta, kiai to ii moyou to ii, nani hitotsu hinan no haire tokoro ga nai, sassoku kaette ousama ni sono mune gonjou itasu de arou"
"Ah! You have woven most beautifully, these colours and patterns, they are beyond any mode of reproach! I shall most humbly convey the information as soon as the King returns!"
A bit more verbose, especially that it is "beyond reproach" and his usage of the humble verb itasu, "to humbly perform/do." His personal pronoun is the impersonal jibun (自分, "one") akin to the English "one" in formal registers, though this usage may merely be that of a reflexive pronoun than a distinctly personal one.
He also calls the Old Minister Daijin (大臣, "Cabinet Minister").
In Kimura's version they are not outright called Swindlers, merely weavers, though they are still are oleaginous:
"Osorenagara mouhiagemasu, watakushidomo wa hataori wo shoubai ni shite oru mono de gozarimasuga nani ka hitotsu orimono no gochuumon wa gozaimasenka"
"Pray allow us to most humbly state that we lowly ones are but weavers who sell our wares, but if you, Sir, would like us to give you one of our clothes?"
Watakushidomo (we humble lot) reeks of self-deprecation, as does gozaimasu (to honourably be), including their usage of gochuumon (example, specimen) prefixed with go to make it extra polite. These decorous words is their mode of speech towards the palace guards - who oddly do not have any lines nor exchanges with the Swindlers in the original, since this is more of an adaptation of the original story than an outright translation.
When petitioning for His Majesty to undress himself, they say:
"De wa, heika ni wa hitatato omeshikae asobashite wa ikaga de gozaimasu, oyobasu nagara watakushidomo ga, okisemoushimashou kara, iza kono kagami no mae ni otachiasobashimase."
"Now then, if Your Majesty would please to deign and change Your Majesty's clothes, then however poor our meagre abilities may be, we shall strive to most humbly doff and garb Your Majesty. If Your Majesty would most graciously stand in front of the mirror and deign to stand still."
If the Swindlers were courteous towards the guards, they are outright kissing and licking the feet of the King. omeshikae (the monarch's change of clothes), asobasu (the monarch does), okisemousu (to most humbly ask to dress) and otachiasobimase (if the monarch most humbly would deign to stand still), whereof two of these uses asobasu, a form of "to do" uniquely reserved for royalty and nobility, here used in the last case with the archaic imperative case asobamase. They also attach oyobasunagara (to the best of one's ability; poor it may be) to watakushidomo to produce a self-effacing request to the King.
Nagura however refers to them with sagishi (詐欺師, "swindler," lit. "masters of swindling") and they too are equally decorous using the same polite register as Kimura has them in their requesting His Majesty to undress. They use other polite forms in:
"Kono orimono wa kumo no su no you ni karuku gozarimasureba, kite mo kita wa omowarenai kamo shiremasen, demo soko ga sunawachi kono orimono no ichiban tokoro de gozarimasuru."
"Since this cloth has the same light qualities as spiderweb, mayhaps Your Majesty is nobly thinking to try these wares on, then the location over there would be the most exalted place to try them on."
Omowarenai the negative form of omowareru (to be thinking), which in turn is also a respectful passive form of the verb, here attached to the King's actions. The Swindlers use the archaic and extra-formal version of gozaimasu, namely gozarimasuru, complete with the conjectural gozarimasureba , which rarely appears outside of highly decorous writing.
The King offers them near the end of the story a rank of nobility which Kimura thought wholly them unworthy of, but Nagura is less of an uncharitable git and translates their rank of Hofjunker (Weaver Knights) as Goyou orimonoshi (御用織物師, "Honourable Weaving Masters").
Lo, His Majesty hath naught on!
The Nagura version renders the revelatory line as:
"Demo ousama wa nanni mo kite inaiwa"
"But the King is not wearing anything"
Rather straightforward, do not that wa is not the same as the emphatic particle but merely the grammatical particle wa here written as ha, but pronounced the same.
"Nanda! Ousama wa nani mo kite iyashinai"
"What! The King is totally not wearing anything"
A bit more forceful, but the point is still communicated across, albeit reduced in terms of Kimura abridging and adapting the story to suit his own version.