On the contrary, it is commonly assumed that much of the tin used in the ancient Near East came from Afghanistan via overland trade. The Old Assyrian texts from Anatolia are explicit about tin being imported from Assyria via donkey caravans (which in turn imported it from further east), and the close association between lapis lazuli and tin in administrative and economic texts from trade hubs like Ebla likewise suggests that much (if not all) of the tin used in Anatolia and Mesopotamia came from Afghanistan.
Additionally, at least some tin was mined locally. For example, the Early Bronze Age Kestel tin mine was discovered in Turkey only about 30 years ago and suggests there's still a lot we don't know about early tin sources.
Although there has been much hubbub about a recent study indicating tin came from Cornwall, there are two issues with this study:
Virtually all of the ingots tested in the study came from shipwrecks in the eastern Mediterranean, with the exception of those found at Mochlos (an island off Crete), and thus the results are overwhelmingly skewed toward tin transported via ships in the Mediterranean.
The study focuses specifically on the Late Bronze Age, and we should be cautious about assuming that Cornwall was a significant source of tin in the Early and Middle Bronze Age.
There have been a few past threads on this question: