I'm currently reading Saul David's The Indian Mutiny, which addresses this question in some detail, as indeed it would have to, given the ostensible importance of the issue in causing dissent among native regiments. In short, it appears that the original cartridges for the Enfields which were sent to India for testing in the climate probably did contain beef and / or pig fat, because nobody had given the matter of religious sensibilities any thought. However, they were never issued, once the problem had been highlighted, but in spite of the subsequent change to sheep fat, as well as the offer for the sepoys to do the greasing themselves, it remained a point of discontent, at least on the surface.
This raises the question of what weapons and cartridges the mutineers used against their former masters. Unless I've missed something, David doesn't address this, though I assume that for the most part at least they were still using smoothbores with ungreased cartridges.
However, one native officer or NCO pointed out to his bewildered white superior that the sepoys had always been keen to transfer to rifle companies of their regiments, and raised absolutely no concern over what was used to grease those cartridges. In short, it was a pretext, along with most of the other ostensible points of dissent.
I'm sure other members have much more knowledge than I, and can comment with greater authority, but the above-referenced book seems to me as good a starting point as any.