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Uranium oxide in the form of the mineral pitchblende was known and mined - as the Wikipedia article says. So the short answer is that as far as anyone in the 1500s was concerned, they were mining pitchblende and using it to make things turn yellow - they had no reason to care about where pitchblende came from.
The use of uranium in its natural oxide form dates back to at least the year 79 CE, when it was used in the Roman Empire to add a yellow color to ceramic glazes. ... Starting in the late Middle Ages, pitchblende was extracted from the Habsburg silver mines in Joachimsthal, Bohemia (now JΓ‘chymov in the Czech Republic), and was used as a coloring agent in the local glassmaking industry. [WP]
Discovering what pitchblende itself was composed of did not come until Klaproth (or possibly, per Wikipedia, until PΓ©ligot in 1841 - Klaproth thought he had identified it, but PΓ©ligot was the first to extract pure uranium).
This situation was not unique to uranium; cobalt ores have been used for millennia to colour things, but the underlying element was not identified until the mid-18th century.