Tl;DR - its Friday night, so beer in hand this might be slightly rambley. But generally, conversion wasn't quick, missionaries largely didn't want settler colonialism, and the idea of a new religion showing up wasn't unheard of.

I'll speak to the Pacific context in general and then just drop some notes about Tahiti as an example- as it was the first island targeted by Protestant missionaries in the late 18th century.

To address the first question- we need to change it a bit, as by and large it wasn't colonists that were converting local populations. Christianity was, in most instances, brought by missionaries from either the US, most prominently the ABCFM (American Board of Commissioners for Foreign Missions- who were also active in converting Native American nations, especially the Cherokee) and the LMS (London Missionary Society). These missionaries were inspired by religious revelation to abandon/sacrifice the lives they lead and cross the globe to bring Christianity to 'pagan' or 'godless' peoples. They were first and foremost concerned with converting local populations- they were not interested in foreign settlement of the islands- and often resisted foreign intrusion. For example, during the Franco-Tahitian War Protestant missionaries like George Pritchard sided firmly with Queen Pomare IV in acquiring firearms from British merchants in resisting French occupation of the island. Missionaries often feared the role that whalers, merchants, or beachcombers would play in corrupting island life, in ruining the good work of the mission. They did not approve of alcohol, sex-work, and the accumulation of capital.

Things begin to change during the 19th century as the geo-political situation in the Pacific also shifts. The children of the missionaries are often less interested in Christian ideals than they are in capitalizing on their position in society. The children of William Henry, an extremely doctrinal LMS missionary in Tahiti, were far less concerned with Christianity than their father had been- none of them ended up active in the church and instead either moved to Australia or engaged in business ventures on the island and were therefor far more obliging to colonial interests (especially markets). As missionaries moved onto more dangerous islands- especially parts of Melanesia- they relied more heavily on state protection and the threat of state retribution if anything befell them and were therefor more open to working with the state when their interests coincided. An example of this would perhaps be Henry Williams and his son Edward who helped translate (or mistranslate) the Treaty of Waitangi from english into Maori- which effectively, as far as the British were concerned, transferred sovereignty of the island into their hands.

So if missionaries weren't generally interested in settler colonialism, especially for the first half of the 19th century, we might answer the interior question of above, by saying, that for the most part populations converted before land expropriation, annexation, or any loss of sovereignty

So how/why did people convert? First, I wouldn't say people converted quickly- often this was a process that took generations to really set it. At first, especially in places like Tahiti, people might convert to Christianity while still continuing local religious traditions at the same time- something that might only fade over time- and might never go away. Many form of Christianity in the Pacific are very localized and have been adapted by local populations to incorporate previously religious symbols and traditions into Christian ones. Again, using Tahiti as an example, the first missionaries arrived in 1796 and while they had some success building a small congregation, it took them until 1815 until the could finally convince King Pomare II to convert- and even then they never really believed he was sincere in his conversion. Still Pomare II made Christianity the official religion of his Kingdom. Yet, a decade late Tahiti saw the emergence of the Mamaia Visionary Hersey; The Mamaia prophets represented a kind of synthesis between traditional Tahitian beliefs, the millenarianism of the missionaries themselves, and their own understanding of scripture- much of which they read in Tahitian translation on their own as there were not enough missionaries to staff all of the churches in Tahiti. The prophets or visionaries informed their followers that the end times had already begun, evil had been banished from the world, thus there was no more hell, no devil, no sin, and no need for laws. The Heresy was obviously not popular with the emerging centralized Kingdom nor with the protestant missionaries and so it was suppressed by the state- though it continued underground and there is evidence it may have had practicing adherents until 1841.

Even then, its not like people completely abandoned local or family gods or spirits or even marea (temples)- some of which have been maintained by families until this day- and the same is true across other Pacific islands as well. As I mentioned above, people had no problem incorporating this new spirituality into their preexisting one.

Part of the reason for this- or an explanation often used- is that people were already used to religious change. For example, followers of the war/fertility God ‘Oro had risen up from the marea at Taputapuatea and conquered Tahiti and many of the leeward islands as well, overthrowing some of the previously paramount Gods and asserting the dominance of ‘Oro around 1500 CE. So the idea of conquest, of imposing a new paramount God was not necessarily all that novel. In fact, the marea at Taputapuatea had been desecrated by warriors from Borabora in 1710, and as he died amongst the desolation, the high priest Vaita went into a trance and delivered a prophecy :

The glorious children of Tetumu, 
will come and see this forest at Taputapuatea. 
Their body is different, our body is different, 
We are one people only from Tetum. 
And this land will be taken by them
 The old rules will be destroyed 
And sacred birds of the land and the sea
 Will also arrive here, will come and lamet 
Over that which this lopped tree has to teach 
They are coming up on a canoe without an outrigger.

So there was a certain belief and faith in prophecy- which foretold the arrival of Europeans or outsiders and new religions. And this was not just present on Tahiti, but is something of a running theme across Pacific islands.

Christianity also had some very real material benefits attached to it- it opened up opportunities for trade and social advancement outside of traditional society.