This is not as a straightforward question as one might think. First and foremost, it is based of a presentistic assumption that the propagation of information back then was as quick and common as today, what is patently not true. Until fairly recently (from historian's perspective at least) the shape of the Earth was more or less a philosophical concept with no practical application and not that different from the proverbial considerations about the 'number of devils capable of dancing on the pinhead' often used to not quite fairly satirize scholasticism.

For the majority of people, 'the world' was quite an abstraction, as most peasants and burghers have seen only the immediate area surrounding their place of residence and extending maybe a dozen miles in each direction. Even nobles who happened to avoid any military expedition could have seen only a territory of that size or maybe slightly bigger. For the ancient or medieval people the shape of the Earth was roughly as abstract as the existence of Earth-like exoplanets was to 19th century merchant. And there are basically no sources stating that this issue was of any concern to an Athenian metic, Carolingian serf or English peasant. On the other hand, many scholars were interested in such ideas, with the Pythagoras proposing the spherical shape of earth in late 6th century BCE, although this was based on philosophical rather than scientific principles. However, it is Aristotle who is usually credited with the first sound arguments for the spherical shape of the earth he made in mid-4th century BCE, with the Eratosthenes of Cyrena following in 250s BCE with the first known calculations of the planet's dimensions that were remarkably accurate (98,77% of the actual Earth's radius).

The prevalence of a spherical Earth idea in the late ancient and early medieval Europe is dubious, but definitely not because of Christianity. The latter became an official religion of the Roman Empire in early 4th century CE and thus the 'Christian world' was quite synonymous with 'Roman Empire' even though the pre-Christian beliefs co-existed with the new religion for some time. Given this political and cultural continuity, the spherical shape of the Earth, sometimes present in the depictions of Roman deities or Emperors as sculptures or on coinage became obvious also to Christians. If this notion was not popular in Europe, it was because large part of the continent (in general, all lands east of Rhine and north of Danube) were either never influenced by the Graeco-Roman world, got settled by people who came from outside this cultural area or devoid of Mediterranean influence that promptly vanished after the crisis within the Roman Empire that started around 3rd century CE. If anything, it were members of the Church who brought the Hellenic thought to these lands, such as e.g. Bede the Venerable, an English Benedictine monk who presented various geophysical and astronomical observations in his book De temporum ratione finished in early 730s. The expression 'orbe terre' (terran sphere) is also used in many sources, e.g. Liber floridus (page 24v) compiled by Lambert of Saint-Omer sometime in the first decade of 12th century. This, as well as a presence of a globus cruciger (cross-bearing globe) signifying power of Christ over the world and commonly used as an attribute of the latter in the Christian imagery or a symbol of temporal royal or imperial power (present on e.g. contemporary depictions of Frederic I) strongly suggest that the spherical shape of the Earth was more or less obvious for anyone interested in such matters.

So, to sum this up, the idea of the spherical Earth was generally well-accepted among the European scholars in Graeco-Roman world since at least 3rd century BCE, spherical depiction of the world are present in numerous artifacts from that region throughout history and was brought along with Christianity to the parts of Europe outside Roman sphere of influence, where it have spread and developed further. It is very plausible that in 14th-15th century, the idea of spherical Earth was obvious, if hardly consequential to anyone with modicum of education.

Columbus knew the Earth was round as did everyone. The problem in backing Columbus' idea were his calculations, as he had estimated the Earth to have a perimeter of 30,000 Km, instead of the actual 40,000. Columbus was not trying to demonstrate that the Earth was round, as that was common knowledge, and had been so for centuries upon centuries.