This question was made for me! Ahaha, it's a very good question. To an extent, we need to go further back. Essentially all pre-Abrahamic religions, and to an extent even the Abrahamic religions, though the degree is a rather hot topic, practiced a form of syncretism.

It was entirely possible to be a Roman Polytheist, a Hellene (Greek polytheist) and have an altar to Isis amongst your tutelary deities, for example.

One of the big changes monotheism brought was an aggressive campaign against interior religious flexibility. Though early Christianity was fairly flexible, and had many different branches that pursued such things as magic and the occult, transcendentalism and unity with the Greek pantheon, these were assiduously weeded out over the course of the centuries - and it did take that much time, with heresies popping up every hundred years or so - sometimes quite successfully.

This is important to your original question, I feel, because almost every indigenous religion attempt to syncretize predominant schools of thought or religion into itself; whether this was the Mahayana branch of Buddhism taking Shinto on a completely different road from the Koshinto all but disappeared from today, or increasingly insular fisherfolk on the White Sea altering old rituals so that they venerated saints instead of propitiating spirits.

Many have suggested that if only the ancient (and not so ancient) religions had been more like the missionary traditions of the time, they might have survived more or less to this day. I would heavily disagree with this, for that is more or less what happened. The longest openly surviving European heathenry, for example, what is now called the 'Romuva' faith - and heavily reconstructed at that - began to adopt many trappings of Roman Catholicism in it's ember years. These only made eventual conversion easier, not the religion 'harder'.

Nor would have attempting to reform or pursue a more dogmatic line worked. The predominant missionary, or prosletyzing, more of the region usually had the upper hand. Raiders could burn monasteries, but they did so infrequently and generally out of a desire to gain wealth, rather then to establish religious hierarchy.

Indeed, I feel if any indigenous religion were to survive largely unaltered, it would need several things.

1): To be more 'true' to itself. This does not mean rejecting foreign influence, as the key unifier of most such religions and beliefs - insomuch as there was one - tended to be a receptiveness, as opposed to dogmatism. Rather, it means establishing a cultural and religious set of standards that are spread throughout the reach of the religion or belief.

2): Literary texts. Not necessarily holy books, but collections of unaltered and unsyncretized stories and beliefs. From Mahayana Buddhism to Christianity, Islam to Confucianism (which itself is a set of social beliefs more then a religion), one unifying factor is strong codified tradition that can be referenced, debated, fought over - but always present.

3): Economic success. This is the hardest, I think, for by nature most such religions and beliefs are tribalistic or ethno-religious. The traditional view of the Norse as conquerer-barbarians comes under fire simply when you look at the number of raids Vikingr went on, versus the number of raids by foreign powers into Norse soil; the same is true of most any such indigenous religion, continuing until they are conquered or convert to a larger religious group out of need for either allies or for the increasing central authority such groupings provide.

So. Taking the example of the Norse, you would probably have to have a completely different history, though I imagine Harald Hadrada (who was by all accounts a very bad Christian, if he even truly was one) winning 1066 might have extended the lifespan of the religion another hundred years, to say nothing of Erik. Assuming the North Sea Trade, or some other network could bring wealth to the followers of the Norse faiths, and they could find common cause - well, the rest is truly a what-if, as we have few examples that succeeded in that matter.

In summary, the typical reaction from indigenous religions tends to be toleration, anger, fear, reaction, and sublimation. And yet; in almost every region today, you can still see the marks of those older religions in most everything from cultural ceremonies to local myths; perhaps that is there way of adapting or reforming, as small a victory as it may seem.

I realize I've rambled quite a bit, but I hope I've nevertheless given some possibility to your question - know it's not the only possibility, as again, even with our best efforts there's certainly a degree of only being able to guess at the answer.