'Work on it' is the correct subject to use as it refers to a specific task or job that is being done. We normally use the verb in the -ing form (gerund) as a subject when we are talking about an activity in general. 'Working too much is bad for you'. Sometimes the noun and the -ing form of the verb are identical and so there would be no need to choose. 'Drilling was started in 2001.'

The phrase 'Working on it " is a so called verbal ( or gerund) phrase and it functions as the subject of the sentence. Working on it ( verbal phrase functioning as SUBJECT) had begun ( predicate) A GERUND phrase will begin with a gerund, an "ing word " ( in this example 'working') and might include other modifiers and/or objects. Gerund phrases always function as nouns, so they will be subjects or subject complements. Another example to show how this sentence is built : " Eating ice cream on a windy day can be a messy experience if you have long, untamed hair." "Eating ice cream on a windy day ' is the gerund phrase , starting with the verbal noun 'eating', it functions as the subject of the sentence ,then comes the predicate "can be " and the object ' a messy experience".

I think technically, they are both correct since you can say "I had begun working on it" and "I had begun work on it", but for some reason, when it's converted to the passive form, "work on it had begun..." sounds much better than "working on it had begun...".

"Work" should be used here. Note that "on it" gives additional information of the noun and can be removed to make sense of the sentences. 1. Work had begun before my sister left. 2. Working had begun before my sister left. Sentence (1) makes perfect sense. Sentence (2) does not make sense. "Working" as a Gerund is the continuous action of doing work. eg. He likes working on his drawings during tea-breaks. In the phrase, "Working hard on it..." working is just a verb in the continuous form and is no a verbal noun (i.e. Gerund). Another example of a Gerund: He likes jogging. Note that for most sentences where gerunds may be used, "to verb" can be used to replace it. eg. He likes to jog. "I/They starting working on it before my sister left." would be the correct use of "working" in the above sentence. "Working" does not function as a gerund here. I hope this is clear.

work (noun)