I notice on the main website there's a page that explains where the punctuation goes is down to whether you use British or American Emglish.
It states in British English, the puctuation goes outside the speech marks. I'm doing some proofreading, and have begun to follow this rule, but it just doesn't look right to me, as a Brit.

I learnt most of my English grammar at primary school (up to age eleven), by the time I got to grammar school, there was hardly any punctuation taught. So it is a long, long, time ago. However, I'm quite willing for someone to confirm it either way.

This can be a convoluted subject, too lengthy to list all possibilities here. For US English, the reference I use is the Webster's Collegiate Dictionary, Eleventh Edition, p 1608, listing a full column on the subject.

Basically, punctuation goes inside the quotes, but not always. Good luck!

Firstly, a comma is used to indicate the separate ideas contained in a sentence.
e.g. "The EU is ultimately doomed, according to many analysts, as the economies of more and more of its smaller members falter."
Note the commas above.
Idea one: The EU is ultimately doomed as the economies of more and more of its smaller members falter."
Idea two: according to many analysts

………………………………………..
Paul : "Give it to me or suffer the consequences."

becomes, say, "Give it to me," said Paul, "or suffer the consequences."
What is INSIDE the quotes is supposed to be a direct quote of Paul's words. So where did the comma after 'me' come from?

It is in the reporting of the Direct Speech that the writer has separated the ideas - and so the comma makes far more sense in separating "give it to me" and "said Paul", as in:

"Give it to me", said Paul, "or suffer the consequences."

I am not changing to an American system that makes no sense to me.

I follow what you're saying about not changing to an American system that makes no sense to you. You've placed the full stop inside the speech marks.

That's the same way I was taught too.

However, the website says that British English should have the full stop outside the speech marks, hence my query in saying it doesn't look right to me.
So do you think this page is wrong? http://www.grammar-monster.com/lessons/quotation_%28speech%29_marks_punctuation_in_or_out.htm

It is not whether the page is right or wrong, since it is merely reflecting the mess this whole punctuation business has become.

I repeat: What is INSIDE the quotes is supposed to be a direct quote of Paul's words. His words end with a full stop:
Paul : "Give it to me or suffer the consequences."

So, when reporting his Direct Speech, I place a Full Stop at the end of his words.

"Give it to me", said Paul, "or suffer the consequences."

Let's make it more emphatic:
"Give it to me", said Paul, "or I'll kill you!."

Now, the page says: Exclamation marks and question marks according to logic.
Logic dictates that it is Paul who is being emphatic, so the exclamation mark goes inside the quotes. It is also logical that Paul's words end in a Full Stop. So a Full Stop goes inside the quotes, making one outside redundant.

Just one more observation: "this whole punctuation business has become" such a shamozzle, continue what we were both taught, and see the logic in it.