(Apologies for any implausibilities here, Classical Greece is not my field of expertise).
Greece falls to Persia, at the same time Phoenicia, for one reason or another, can no longer bar the Straights of Gibraltar. Existing colonies expand and new ones are founded rapidly, including some on the Atlantic coast (not very sure on this, Mediterranean ships often struggled in the Atlantic, perhaps they adapt over generations, or learn from the locals). More Greeks are involved in North Sea commerce, including some who decide to go to this mysterious place where Amber comes from, eventually, Iceland is discovered by traders blown off course or (probably more likely) a Strabo-type expedition. This is probably about the last place on Earth most Greeks would fathom fleeing to, maybe a fringe religious movement in this thoroughly-altered world? They might not be especially Greek anymore, but I figure you need them on the same side of the continent for this to be conceivable.
Alternatively, A Greek state takes the place of Rome, attaining a similar size, most importantly, including Britannia, my inclination is that even with a Greek *Roman Empire, the more nautically-inclined Greeks explore further than Rome did, including into the North Atlantic.