The fruits don't exactly travel between different climates season by season, but they are quite capable of traveling on their own.
Coconuts come from one of several plants that spread themselves using drift seeds, which can travel for thousands of miles over by floating in the ocean until they land on distant shores. If a coconut falls into the ocean, its buoyant husk allows it to travel wherever the ocean currents lead it.
The question remains: could a coconut possibly migrate from its native tropical climate to a more temperate region, such as (and this is just off the top of my head, mind you) the Medieval Anglo-Saxon kingdom of Mercia, located in modern-day England?
To answer that, we need to take a look at where coconuts grow, and how ocean currents work.
There, does that clarify everything?
Ocean currents have been known to spread debris to locations far away from its point of origin. One noteworthy example comes from 1992, when a shipping crate containing 28,000 rubber ducks was lost at sea near Hong Kong. The ducks traveled across the currents for years, washing up on the shores of Australia, Argentina, Alaska, Washington state, New England, and eventually the east coast of England itself.
After falling off the ship, the rubber ducks were swept along by the Kuroshio Current, a current which also passes by the Philippines. Since coconuts have grown in the Philippines for millenia, a medieval Filipino coconut could well have followed the same oceanic path as the rubber ducks from Hong Kong.
The kingdom of Mercia included small sections of England's east coast, so it's possible that some wayward coconuts could have been swept to the kingdom's shores and picked up by a travelling king with aspirations of uniting the land.
However, he'd likely have no idea how the coconuts got there, which is a shame. You have to know these things when you're a king, you know.