Screenshots from two different games, happened in a third too just don't have the screenshot.
The Saudis (Nejd) regularly import so many slaves that some 40% of their population is slaves, after which they revolt and gain their independence from their masters, resulting in an African nation in Arabia.
At present in game outside of a few rare circumstances it is not possible to change state religion. However, there is one not so rare circumstance which is not reflected, which is that of an unrecognised nation adopting the religion of a great power, often for diplomatic/political reasons.
The Precedent
To me the most prominent of precedent for this comes perhaps from the Tswana, who are admittedly a decentralised nation in game. The Tswana people would become subjects of Britain through the Bechuanaland Protectorate, formally to protect them from the Boers. Khama III in particular could speak Dutch, got baptised as a Lutheran Christian, and had his family and the majority of his tribe follow him in this conversion, making him the founder of a Christian state of which he became king. Even though this state did become a colony of Britain, he clearly got more recognition of his rule and autonomy, the protectorate remained independent of Rhodesia or South Africa, and ultimately his grandson Seretse Khama studied at Oxford, married a British woman, and returned to become the first president of an independent Botswana.
For a different example the results of which are visible in game, Hawaii had just recently become a protestant Christian country (and adopted a British-inspired flag) under Kamehameha II, which also came with the abolition of old traditions and the institution of a new legal code.
Ranavalona II of Madagascar similarly converted to Christianity along with her court in 1869 and burned old idols in the process of this public conversion. This is perhaps the most egregious individual omission, considering Madagascar being a playable country in game which starts out as animist.
Importance
Conversion to Christianity was often something that allowed rulers to do away with old traditions, consolidate power, modernise their countries, and receive foreign support and recognition. Even when colonised, Christianised tribes were seen as considerably more civilized and they often received better treatment or greater autonomy. Ultimately, it is also especially clear that "paganism" would not be taken seriously in the world and was largely seen as primitive superstition synonymous with tribalism, while "religion" was increasingly a term encompassing "world religions" which carried considerably more respect. That development and organised religions tend to go hand in hand historically is probably also worth at least some note here.
Ethiopia being Christian to begin with also increased their legitimacy and prestige, and this is also not reflected in game at present.
Victoria 3 is ultimately a game which allows the player to pursue an alternate history where such modernisation and recognition succeeds even where it may have failed historically. As a result, it's important for the option to be on the table.
Mechanics
In Griqualand, the devout are already styled as the "London Missionary Society" and have the laissez-faire ideology, aiding in modernisation. They also generally have foreign IG leaders. This provides a basic model for how this sort of change could work in game in terms of a foreign missionary devout.
The actual conversion should probably work through some sort of journal entry/decision and should probably require contact and interest from a European/Christian country. The choice of specific denomination should be dependent on having contact with the relevant kind of country, and should make for better relations with countries of that denomination especially.
The existence of missionary societies should open up diplomatic options which aid in improving relations and in modernising your country with the help of foreigners. Generally a protestant country might be seeking aid from Britain while a Catholic one from the French, and may have devout IG leaders from these countries.
In addition to technological development, the conversion and the new Christian devout should help with passing more modern laws and with opening up the country. It should also be a double edged sword in pushing for policies like free trade and easily pulling a country into a power bloc or making them a victim of economic or direct imperialism.
The recognition journal entry should punish animism, decreasing progress for being animist and making it more difficult to get recognised if not following an organised ("civilized") religion.
Conversion should be one time and convert some fraction of your population immediately. It or the resulting discrimination or modernisation could also potentially cause a civil war with animist reactionaries who see the new religion as "colonial"
Other religions?
The choice should certainly only be open to animists, countries with organised religions would be too entrenched in their ways to make a radical change, and there's not really precedent for it either. (though missionary activity did take place in Asia so it could be argued that it should be more difficult but still technically possible in some other way)
However, the options of what religions to convert to are a more interesting question.
Christianity is the obvious historical option, there's not a lot of conversion to anything else in this time period. However, player choice and historical circumstances could still be taken into account.
One option is to allow conversion to nearby world religions, so for instance animists in Asia could also convert to (forms of) Buddhism or Africans might be able to convert to forms of (Islam). Another option might be to make it so it is possible to convert to the religion of any great power, or any great power with an interest in the region.
Rewards should probably be related to this. In terms of recognition, perhaps only animism would offer a straight up malus, but a bonus could be offered by any religion (or religion group) which has a great power. So for instance if you convert to Islam, but the Ottomans lose their status, then you would also lose the bonus to gaining recognition. By contrast if Japan becomes a recognised great power, it could make Buddhist states more legitimate in the eyes of the international community.
Similarly, relations with and bonuses from a great power are only possible while that great power exists and has an interest in your region. As such major colonial powers would basically always make most sense to align with, even if the player might have other options for roleplay purposes.
Conclusion
The religious aspect of colonial politics, international relations and modernisation is almost entirely ignored by the game at present and it would not really take much more than a generic journal entry and some choices for any animist country to reflect this and make for a much more interesting and dynamic modernisation process for some of the least developed countries in the game, while also providing representation of the influential missionaries of the time.