It was a recording of a children's story on a cassette tape.  The
publisher was British, so I assume it's a British author, but I do not
recall any specific names.

I also don't know how the story begins, but I know that the young boy
was out fishing, and he caught some type of talking fish.  The fish
magically produced a rainbow in his pail, and the boy returned home
with it after thanking the fish (the fish being obviously gracious
because he wasn't eaten).

Along the way, the boy finds an injured person who is cut somewhere, I
think on the leg.  The boy bandages a the cut with a piece of the
rainbow.  He also enounters a hungry person, to whome he gives another
piece of the rainbow.  There are a couple of additional stops along
the way before the boy reaches his house.  He looks into his pail, and
sees that only a small piece of the rainbow remains.  He is not
perturbed by this; rather, he is satisfied that he helped so many
people on his way home, and the end of the tale finds him content.

I'm trying to find the title, author of this story and whether it's
available in a print edition or in cassette tape. Thanks in advance
for any help!

Could it be "The Last Slice of Rainbow"?
http://www.amazon.com/exec/obidos/tg/detail/-/0060200421/qid=1128623721/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/102-0661294-1788964?v=glance&s=books&n=507846

"The Last Slice of Rainbow and Other Stories" does sound like the one.

"While upper elementary children can enjoy these contemporary fairy
tales independently, a good storyteller could present them to any
group from third grade through adult. Each story creates vivid images
that return to readers as they discover what the story has revealed
about human nature. The title story is about a boy who acquires a
rainbow but discovers that being a person who can love rainbows also
means being a person who has to share what he loves..."

http://www.amazon.com/exec/obidos/tg/detail/-/1856561003/103-5106196-1956626

Best regards,
Rainbow

Thanks, I've just ordered a copy. Hopefully, it's what I'm looking for...

Hopefully it is the right. Please let us know.

All the best,
Rainbow