From 100 proofs the Earth is not a globe on the TFES wiki:
2. Whenever experiments have been tried on the surface of standing water, this surface has always been found to be level. If the Earth were a globe, the surface of all standing water would be convex. This is an experimental proof that Earth is not a globe.
The answer to this was given prior the claim man made it to space.
8. If the Earth were a globe, a small model globe would be the very best - because the truest - thing for the. navigator to take to sea with him. But such a thing as that is not known: with such a toy as a guide, the mariner would wreck his ship, of a certainty!, This is a proof that Earth is not a globe.
People have been getting to where they need to go with maps created based on the assumption the world was round prior to the 1960's.
Claudius Ptolemaeus A.D. 85-165). His "world map" depicted the Old World from about 60°N to 30°S latitudes. He wrote a monumental work, Guide to Geography (Geographike hyphygesis), which remained an authoritative reference on world geography until the Renaissance.
The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for nautical purposes because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments that conserve the angles with the meridians.
9. As mariners take to sea with them charts constructed as though the sea were a level surface, however these charts may err as to the true form of this level surface taken as a whole, it is clear, as they find them answer their purpose tolerably well - and only tolerably for many ships are wrecked owing to the error of which we speak - that the surface of the sea is as it is taken to be, whether the captain of the ship "supposes" the Earth to be a globe or anything else. Thus, then, we draw, from the common system of "plane sailing," a practical proof that Earth is not a globe.
See 8 above
This is suggesting at least a large percentage of ship wrecks were the result of maps being based on a round Earth. Would some of the reasons for the wrecks have to be covered up?
10. That the mariners' compass points north and south at the same time is a fact as indisputable as that two and two makes four; but that this would be impossible if the thing, were placed on a globe with "north" and "south' at the centre of opposite hemispheres is a fact that does not figure in the school-books, though very easily seen: and it requires no lengthy train of reasoning to bring out of it a pointed proof that the Earth is not a globe.
Lets assume 1885 was when this information was decided to be withheld from the general populace.
17. Human beings require a surface on which to live that, in its general character, shall be LEVEL; and since the Omniscient Creator must have been perfectly acquainted with the requirements of His creatures, it follows that, being an All-wise Creator, He has met them thoroughly. This is a theological proof that the Earth is not a globe.
Hard one to get a start date for. Lets assume some time after 300 BC some people figured out and proved several Greeks made wrong conclusions from their observations of the world around them and solar system.
18. The best possessions of man are his senses; and, when he uses them all, he will not be deceived in his survey of nature. It is only when some one faculty or other is neglected or abused that he is deluded. Every man in full command of his senses knows that a level surface is a flat or horizontal one; but astronomers tell us that the true level is the curved surface of a globe! They know that man requires a level surface on which to live, so they give him one in name which is not one in fact! Since this is the best that astronomers, with their theoretical science, can do for their fellow creatures - deceive them - it is clear that things are not as they say they are; and, in short, it is a proof that Earth is not a globe.
Similar to 17 above
At some point in history someone had to decide to withhold discoveries. As stated above it should be simple to prove the Earth is flat. Unless you believe throughout history such a simple thing to prove was never done by anyone else. So again we are looking at some point in time after 300 BC.
Just curious why focus on the space travel? The above is a small sample from the TFES wiki that suggest that the Conspiracy may have started prior to 1885. The proofs above suggest some or all people involved with cartography and general science may have been withholding information the Earth was flat.