Conversion to Christianity was not the only option women had. The purpose of the 1939 Act was to “consolidate and clarify” provisions already available to Muslim women. Contemporary writer Sheikh Abdul Haq pointed out the use of conversion to obtain a divorce had two issues. First, that it was a misinterpretation of Muslim law by non-Muslim courts. Second, that it was the result of women having almost no other options.
From the book Inscribing South Asian Muslim Women: An Annotated Bibliography & Research Guide by Tahera Aftab:
He goes on to argue that the traditional Islamic law should be restored, which would grant women more leverage.
‘Tafwiz’ refers to “talaq-i-tafwiz” which is the right of a Muslim woman to dissolve her marriage on her husband's behalf. This option for divorce was delegated to a wife if her husband chose to do so.
From Talaq-i-Tafwid: The Muslim Woman’s Contractual Access to Divorce: An Information Kit
This publication primarily focuses upon a procedure recognized by
Islam – talaq-i-tafwid or the contractual delegation to the wife of
the right to pronounce talaq on behalf of her husband in regard to her
From the same publication there are two excerpts from pre-1939:
“Delegated divorce, talaq-i-tafwiz. This is a curious survival of the
old property rights of the husband over his wife. He may appoint an
attorney to divorce her on his behalf just as he might appoint an
attorney to deal with his property.”
Seymour Vesey-Fitzgerald, Muhammadan Law: An Abridgment According to its Various Schools(Oxford: Oxford University Press, 1931), p. 77.
Delegation of divorce is the assignment of a power . . ; it is not
agency, and therefore the husband can not retract from it; so much so
that if he gives her . . . [such authority], and then swears that he
will not repudiate her, but she divorces herself, according to the
most approved opinion, the husband shall not be treated as guilty of
breaking his oath
Muhammad Ala-ud-Din Haskafi, Durr-ul-Mukhtar (trans. by B.M. Dayal; 1913 edn. reprinted Law Publishing Co. Lahore, n.d.), p. 172
In addition I have found the term khula, mentioned in this thesis:
I didn't find many useful references to this term, but in concept it seems similar to Tafwid. 'Mehr' is a kind of dower, paid by the groom to the bride. Women who initiated divorce often lost the right to their own dowry as well. In many of the references to Tafwid, I found that it was primarily upper class or intellectuals who were allowed to use (or knew of) this method for divorce.
If using conversions was due to courts misinterpreting Islamic law, the chances of an uneducated woman knowing her options were slim. For those who had no means of navigating a divorce, conversion may have been the least strenuous option. The question also asks how common this was, which is difficult to estimate given that India is a huge and diverse country, and laws can be interpreted differently from place to place.