Saul David devotes some time to this question in Victoria's Wars. In short, his conclusion is that the issue of cartridges was more of an excuse or a rallying cry than a real issue. The rebellion was more driven by other factors, particularly a general loss of prestige and status.

It's not entirely clear, but it seems that the test cartridges prepared for use in India did originally use a mixture of animal tallows. These were prepared by the Bengal Ordnance Department against a warning from the Commander in Chief that they should not be "of a nature to offend or interfere with the prejudices of caste."

When the rumour spread that the cartridges were greased with pork and beef tallow, planned firing drills were altered to allow sepoys to apply their own grease. Despite this, a regiment at Bernhapur refused to drill with the new cartridges.

Objections "it seems, had switched from the grease on the enfield cartridge to the paper used for the new cartridge, and finally to the paper on the old musket cartridge." The British, not unreasonably, suspected the issue was being manipulated by agitators. In response they changed the drill so that sepoys would tear cartridges with their left hand rather than their teeth.

If cartridges were not the main issue, what was? Sepoys had quite a range of grievances and motivations for rebellion, including:

• A falling proportion of high caste hindus in the ranks.
• Declining rates of pay.
• An end to the practice of only taking volunteers on overseas service (dying overseas could lead to a person losing their caste)
• Reduced opportunities for making money from plunder, without serving overseas.
• Poor opportunities for power and promotion, and being commanded by younger, less experienced, but white, officers.
• Centralised control reducing the ability of officers to discipline their men.
• The annexations of Indian states by force or cunning, particularly that of Oudh in 1856, an area which supplied as many as 3/4 of sepoys.
• The support of many dispossessed (but often still wealthy) Indian princes, nobles, officials, etc.

So while the introduction of the new cartridges seems to have genuinely been viewed badly, it was exploited as an issue. Discontent had been reported before the new cartridges, and the rebellion itself didn't break out for some time after their introduction, and after a range of concessions had been made by the British.