I'm not any sort of authority on the matter, but I think there's a separate word for when - कब. If using "when", an answer might be "at night" or "later today" or "when he's done working", however "at what time" can only ever result in a numeric answer. So while I agree that "When" is a more natural equivalent in many cases, it lacks the same level of precision about what's being asked.
A native speaker of Urdu here, trying to learn Hindi script.
This sentence cannot be literally translated in English as it appears and needs some explanation. In fact this verb does not exist in English and all the confusion here is precisely because of that.
BAJNA is a verb that roughly means... to strike, to ring spontaneously (for example a tone of a bell), to play spontaneously (for example a song on the radio). It is something akin to a reflexive verb in French. A song on the radio is BAJTA HE. A bell can do the same. This verb seems to explain the reflexive action of a clock when it sounds the hours by striking the mechanism inside.
So ITNE BAJE being in the past, can be translated as...HAVING STRUCK SO MANY TIMES...
I am not sure but I postulate that the verb entered the lexicon after the clock was introduced to the subcontinent. When it struck at a given hour, it was doing the action of BAJNA and so many time depending on the hour.
How many time the bell rang....KITNE BAJE (How many strikes?) This many times .... ITNE BAJE That many times...UTNE BAJE etc.etc.