I have named my fish and dolphin characters in my short story. It transpires that these names are already in use in other stories, for example 'Cornelius Cod'.
Would I have to change my characters names?
I live in England.
There are two separate issues here:
For the first, I don't know the laws in the UK (though laws in other countries may apply if the book is published there). In the US, it would be an issue if the name were trademarked (which is something that costs a lot of money and is not the same as copyright). So you could not name your character "Charlie the Tuna" or "Morris the Cat," for example.
For the second, yeah, I would. Cornelius isn't super common but it's well known enough that I'd keep it. But Cornelius Cod is quite specific and keeping it would be confusing to your readers familiar with the first Cornelius Cod (they'd expect it to be the same character). It would also reflect poorly on you. I realize this isn't the name you're using, but I can only comment on the example I have.
I agree with Stephane that you can use more iconic names (Tarzan being a great example) when they're given to characters by other characters, as nicknames. You could also make puns or plays on words, like "Martin Luther Kingfish."
But I absolutely would not duplicate a complex name someone else has already used. Nor would I use simple names that are already strongly identified with a famous fish or other aquatic animal (Nemo, Flipper, Shamu), unless it's used as above.
You do not have to change those names if it makes sense in your story to use names from other characters, fictional or historical. What you cannot do is to use those as if you were the first one.
Imagine you want to name one of your character "Tarzan". You must justify the use of such name by some explanations, like this :
I didn't know his real name. We used to call him "Tarzan". Actually, he was tall, strong, swimming like an alligator. And he was as clever as Chita, his sister.
If it does not make sense in a way or other, you will be regarded as a lazy writer and your readers will read you work suspiciously, it they do.
I would recommend choosing a slightly different name for each. As others have said, it might be confusing for your readers to see the same name in different series. If the characters are mostly different apart from the name, changing the name should prevent any legal issues. I could see issues being caused by the character being too similar to the character with the shared name, because then a name change could be seen as, "well I changed the name! Clearly it's not the same character," when the character is the same in every way apart from the name. It would be like making a character that is exactly the same as Harry Potter and naming them something else. But that doesn't seem like the case here, so I'm sure you don't have to worry about that.