If Marcion got his way, the canon would be extremely restricted as he only supported Luke-Acts, and some of the letters of Paul. I think a more likely scenario would be for a watered down version of Marcionism to take hold with the Gentile converts to the faith. Perhaps have more Gentiles than Jews in the early church would do it.
A lot of the canon was agreed upon in those first few centuries although some books were much more controversial than others. Eusebius of Caesarea in his Church History (4th century) doubted whether Revelation should be part of the canon, for example. So I can definitely see more Jewish parts of the NT being dropped.
Christian theology without the OT would be different, even more Hellenized than before. I could see this alt-Christianity being more likely to turn and twist with the fashions of thought in whatever particular age it found itself in. Sure, Platonism and Aristotelianism have dominated Christian intellectuals in various periods of history but without the OT, there would be less conservative ballast in the faith to make it similar across time periods. Dualism would dominate as the strong notion of the creator-God who created this world and said it was "very good" would be replaced by a distant God or perhaps just a higher god in the Great Chain of Being. Jesus would be seen as one who came to teach an austere ethic before being killed by the evil demiurge that created this world. The resurrection of Jesus would be reinterpreted into just a higher spiritual state absent of physicality rather than its Jewish root of being something that happened bodily. Greek thought at that time thought resurrection-talk was ridiculous just as much as many moderns scoff at the idea.
Paganism wasn't very hospitable to the scientific method in that the myth of eternal return doesn't breed curiosity and the will to discover. If all life is cyclical from the earliest of ages then there is no reason to pursue 'progress.' The Jewish idea of linear history might have helped develop modern science along with the foundational belief that this world was created good and meant for human discovery. Alt-Christianity would be even more of a wet blanket on science and technology than paganism as it is all about the 'secret knowledge' and not about serving God in this present world. Newton and Bacon would have existed in a world that was very much distrustful of the natural world, viewing it not as the place in which God lovingly created but a place one needs to escape from. Souls in fleshly cages, to rephrase a Platonic idea.
This is assuming that alt-Christianity survives that long. It may just be subsumed into a general paganism. This is perhaps the most likely as Christianity had many enemies from day one. Reading Celsus (2nd century) against Christianity reminds me of reading Nietzsche or even the atheists of our own day. The OT historically helped Christianity survive and thrive apologetically as it gave the apologists room to argue for fulfilled prophecy and the ancientness of Christianity. (Back then being ancient for a religion gave it prestige while being the new kid on the block detracted from it.) Without a strong Jewish core, alt-Christianity would have sunk into the fields of current pagan thought. It would have less will to resist such a slide since it already jettisoned a huge part of what it was originally all about.
This is all at the height of conjecture of course. I love ancient history but extrapolating how ancient changes would impact things many centuries down the line is guess-work at best given the pesky butterflies.