In 1849 Prince Clarence, son of the king of the Mosquito Coast (now
Zelaya department, Nicaragua)was being educated in England when he
died aged 14 and was buried in the churchyard at Moreton, Dorset. Why
and where was he at school and what was his connection with Moreton?

Since this is largely speculative, I am posting as a comment rather
than an answer and leaving the question open to other researchers who
may have better luck or research skills.  I could not find all the
information you wanted.  But when you become a little familiar with
the history of the time ( I have given a brief summary below), it
could be that the prince was in England for two very good reasons, one
of which would be personal safety.  The other could very well be that
since England was restoring its own influence in the Miskito region,
they wanted a prince and future ruler who was already trained to be
"English friendly" in customs and philosophy.

# 2) Prince William Clarence*. b. 1833 (s/o  Queen Juliana), educ. in
Belize and in England. He d. in England, 1848. - Birth and Death dates
from "Mosquitos, a brief history"
( http://www.4dw.net/royalark/Nicaragua/mosquito2.htm ) - With the
above dates (being only a year in difference from the date and age
mentioned in your question)  it would be highly doubtful we are
speaking of two different princes.

During the 17th century, thousands of English settled along the
eastern coast of Central America. The area extended from Campeche in
Mexico to Panama. The colonists allied themselves with a few of the
local tribes including the Miskito.  These tribes formed a barrier
between themselves and over a million spanish colonists.  In 1687, the
first King of Mosquitia was crowned under English urging.  In fact,
that English urging was so strong that Oldman, King of the Miskito and
son of an unknown Miskito King was taken to
England where King Charles I presented him with a crown and other
items which eventually formed part of the Miskito royal regalia.

In 1740, the British declared a protectorate. This sparked an
additional set of clashes during the near endless British-Spanish
warfare of the eighteenth century. During the American Revolution, the
British were expelled. Terms included in the Treaty of Paris in 1783
called for the Mosquito Coast settlers to withdraw to Belize.  Soon
after, the Miskitos were brought under subjection by the Spanish army.

After the collapse of the Spanish Empire and Central America was
divided into mutually distrustful states, Britain regained a lot of
its influence. In the aftermath of the U.S.-Mexican War, the British
became worried that its protectorate might fall to the Americans. 
British troops were sent to occupy the Mosquito Coast, and expelled
the last Nicaraguans in February, 1848, the same year Prince William
Clarence died.   Britain continued to occupy the Coast through the
1850s, with a brief interruption by Henry Kinney (an American) at
Greytown. It was only in 1860 that Great Britain signed a treaty with
Nicaragua giving up all claims to the Mosquito Coast.

I also searched the schools in the Dorset County which have websites
and were in existence in the mid 19th century.  In a search of famous
students at those schools, I did not find a Prince William Clarence,
nor did a search of Dorset area cemeteries come up with a grave.

Another possibility has to do with some historical confusion:

"    *  1) H.M. George* Augustus Frederic II, King of the Mosquito
Nation (s/o Queen Juliana) - see below.
* 2) Prince William Clarence*. b. 1833 (s/o Queen Juliana), educ. in
Belize and in England. He d. in England, 1848." - This quote is from
the same website kinked above and in fact the first part of the quote
is also used above.

However, notice the names of the princes are followed by asterisks. 
That asterisk leads to the following:
"*these two princes have been confused as a single individual as a
consequence of an error in the writings of E.G. Squier, who omitted a
comma when listing the King's sons in his testimony. "…the Princes
George, William Clarence and Alexander…" being entered as "… the
Princes George William Clarence and Alexander…". He then continued the
error in all his subsequent writings, along with a number of other
mistakes on dates and people, confusion (confusing) many others who
followed." - From the same website, brackets mine.

I hope another researcher can pick up on this and bring the answer to
some kind of conclusion.  But as for me, I have run out of leads.  For
now, anyway.

Cheers
digsalot

The only further lead I can find, apart from the information given by
Digsalot, is from an article by Michael Olien from the University of
Georgia published on the Mosquito rulers. Again, it does not provide a
full answer to your question, so I am placing it as a comment rather
than as a paid answer.

From it, two possible reasons emerge for the prince to receive an
education in England.  The first is that this gave him greater
credibility among the Miskitos.  The second is that Colonel Alexander
MacDonald, then superintendent of Belize wanted the prince to become
king in preference to his elder brother, because this would extend the
period during which he (MacDonald) could influence Miskito politics.
It is not unreasonable to assume that MacDonald thought a spell in
England would make William Clarence feel more favourable to the
British.

Olie’s article cites a source that says MacDonald sent the boy to
England in 1842 with his wife, to be educated in Fulham, London. 
Since the boy was 8 years old at the time, the likelihood is that he
was sent to a small preparatory school, which would not necessarily be
well known. I could not find information on where he continued his
education.  I also cannot find a Dorset connection.  One possibility
is that the MacDonalds had a house or relatives in Moreton and that
the boy died there while on a visit during the holidays.

 “At least as early as 1687, the Miskito believed that in order for an
individual to legitimatize his claim as king, he must be recognized as
the group's leader by the English. Several of the kings actually went
to England, others received commission from the governr of Jamaica and
later from the English representative in Belize. In addition, some of
the princes were taken to England, Jamaica, or Belize to be educated.”

Olien recounts how Colonel Alexander Macdonald, the British
superintendent of Belize took a personal interest in the affairs of
the Mosquito coast.  He wanted to ensure that Robert Charles
Frederic's youngest son Clarence would become the next king, because
that would extend the period over which he could influence Miskito
politics. WS Murphy, a US special agent wrote to Daniel Webster, the
Secretary of State on January 20, 1842 about a visit to Macdonald in
Belize:
“His Excellency shewed me, at the same time, a little native Musqueto
Boy, about 8 or 10 years of age, whom [sic] he said was the Son of the
"Present King" of the Musqueto Kingdom, and Heir to the Throne. That
although the King, his father, was not old, yet he was very and could
not live long, and that at his death, this Boy, would ascent the
Throne of that Kingdom. He said, he had the Boy under his care, and
was to send him to England to be Educated”

[Olien  then refers to the error of Squier in mixing together the
names of the two princes, as described above]

Macdonald wanted the English government to pay for the education of
William Clarence in England, but this was turned down and Macdonald
was told to arrange for schooling in Belize or Jamaica, using funds
allocated by the English for presents to the Miskito Indians (Burdon
1935:63). Macdonald then decided to send his wife and Clarence to
England at his expense, so that the boy could be educated at Fulham
(Burdon 1935:64, 75). [Olie’s reference is to Burdon, J.A., ed., 1935,
Archives of British Honduras, vol. 3, From 1841 to 1884. London:
Sifton Praed and Co].

However, although Macdonald had succeeded in getting the King to
recognise Clarence as his heir, after the King’s death George Augustus
Frederic, the eldest son, was chosen heir and crowned at Belize in
1845, while William Clarence was in England.

The reference to the death of William Clarence in England in 1848
comes from: Anonymous, 1849, Mosquito, Nicaragua, and Costa Rica.
London: Effingham Wilson

From: The Kings And The Lines Of Succession by Michael D. Olien,
Department of Anthropology
University of Georgia, Athens, GA 30602  Published in Journal of
Anthropological Research  39:198-241 [1983]
http://anthro.dac.uga.edu/research/miskito/KINGS.htm