First, by the Middle Bronze Age, there were extensive trade routes running all over the old world, from Britain to Africa to Central Europe to the Near East to central Asia. The inter-connectivity of the Middle and Late Bronze Age would not be witnessed again until the Roman Principate, more than a millennium later. The most common source of tin in the Bronze Age was cassiterite, SnO2, which deposits in alluvial material and is relatively common on or near the surface in tin-rich areas. There were no significant sources of tin in Greece, though isolated pockets might have been found and removed through the centuries. The major sources by the Mycenaean period were, as you mention, the British Isles, Iberia, France, Germany, and central Asia see here. It has also long been assumed that there must have been significant tin deposits in Turkey, but those locations are not known (this is a geological/geographical argument about which I know very little). Tin reached the Mediterranean via extended trade routes, often passing through several intermediaries (it must have been extremely expensive by the time it reached your Mycenaean smith). It has long been assumed that Iberia was a major source of tin for the Med., both from its own local sources and as an intermediary for the British Isles. Some interesting modern studies have been done on extant tin objects from various sites of the LBA, with results indicating that tin came from all over the place, and from much farther than we might think. We find similar results when we do isotope analysis on lead objects from all ancient periods. Metals "got around." Some LBA tin from Israel has been traced to the British Isles. That does not mean that one ship left Cornwall and sailed all the way to the Levant. It could have taken that tin years to make its way, through countless intermediaries (just as a caveat).

A note on your Mycenaean smith: he most likely did not source his metals himself. As a Mycenaean he was living under a palace economy, in which the local ruler stockpiled resources at his palace (like that at Mycenaea, or Tiryns, or Athens, or Knossos) and then redistributed those resources to his populace. Many of the Mycenaean Linear-B tablets that survive involve the reception, distribution, and tallying of such resources, including metals.