Kansai-daialect: なんや よう わからん うちに気を うしのうて しまったが...
Standard Japanese: なんだか よく わからない うちに気を うしなって しまったが...
Although I fainted myself while I wasn't aware of it better ...
I'm sorry I don't know the precise concept or rules behind the transformings.
EDIT
In the explanation here, the example of the questioner can be read as if it is the relationship between ウ[音便]{おんびん} in the Kansai dialect and 促音便{そくおんびん} in the standard Japanese, but as shown below, there are examples conforming to this rule and examples not conforming to this rule.
Looking at the examples listed, even if a word in standard Japanse of the past tense form has the same sound as the word to that the rule can be applied, it is understood that this rule cannot be applied unless the ending form of the present tense of the word is "う".
現在形 促音便 ウ音便
Standard Kansai
Japanese dialect
失{うしな}う 失{うしな}った 失{うしの}うた
会う 会{あ}った 会{お}うた
合う 合{あ}った 合{お}うた
有る 有{あ}った 有{お}うた
仕舞う 仕舞{しま}った 仕舞{しも}うた 《片付けた》
終う 終{しま}った 終{しも}うた 《片付けた》
しまう しまった しもうた 《しくじった》
買う 買{か}った 買{こ}うた
勝つ 勝{か}った [勝]{こ}うた
擦る 擦{す}った 擦{す}うた 《使い果たす》
吸う 吸{す}った 吸{す}うた
縫う 縫{ぬ}った 縫{ぬ}うた
塗る 塗{ぬ}った 塗{ぬ}うた
叶う 叶{かな}った 叶{かの}うた
適う 適{かな}った 適{かの}うた
這う 這{は}った 這{ほ}うた
貼る 貼{は}った 貼{ほ}うた
張る 張{は}った 張{ほ}うた
舞う 舞{ま}った 舞{も}うた 《踊{おど}った》
待つ 待{ま}った 待{も}うた
違う 違{ちが}った 違{ちご}うた
庇う 庇{かば}った 庇{かぼ}うた
EDIT 2
Explaining the relationship between "失{うしの}うて" and "失{うしな}って" by imitating the explanation here it will be as follows.
"失{うしな}う" is a "ワ行{ぎょう}五段{ごだん}活用{かつよう}動詞{どうし} w-sound godan conjugation verb", so it has a stem ending with the "[渡り音]{わたりおん} glide" or "半母音{はんぼいん} semivowel" with "w": "ushinaw-", and underwent different developments in "関西地方{かんさいちほう} Kansai region" and "関東地方{かんとうちほう} Kanto region" regarding "語尾変化{ごびへんか} inflection". In Kanto region, the "w" was interpreted as a consonant, and was used to trigger gemination (a.k.a. 促音便{そくおんびん}) in 連用形{れんようけい} or in past tense:
ushinaw-te → ushinatte 失{うしな}って; ushinaw-ta → ushinatta 失{うしな}った.
On the other hand, in Kansai region, the "w" was turned into "u" (high back vowel), and then underwent further change of vowels (k.a. ウ音便{おんびん}):
ushinaw-te → ushinaute → ushinoute 失{うしの}うて; ushinaw-ta → ushinauta → ushinouta 失{うしの}うた.
In addition, the dialect used in the Kanto region has been regarded as the standard Japanese language. Wikipedia says here "Edo (now Tokyo) developed into the largest city in Japan, and the Edo-area dialect became standard Japanese."
If you read the above explanation carefully, you can see the reason why only the verbs ending with "う", that is, "ワ行五段活用動詞 w-sound godan conjugation verbs" conform to the 促音便 and ウ音便 rule, among the examples I mentioned in EDIT.
うしのうてしまったが
-- へえ・・漫画のセリフですかね?「うしなって」は「うしのうて」になってるのに、「しまった」はそのままなのね・・