Note that your BBC link does use advice as an uncountable:
The websites and helplines below can offer help and advice on how to stay safe and how to get access to emergency refuge accommodation. They can also offer advice if you are worried about the safety of someone close to you.
Not "an advice" but "advice". Note that help is also uncountable here.
The page does mention:
... an advice line ...
Here it is the line which is countable, not the advice.
Your second source does use advice as if it were countable:
Our latest advices from Santo Domingo state that the Spanish troops have almost entirely abandoned the island.
... but note that it is from a correspondent in a Spanish colony, and the text dates from 1865. It is not considered normal to use advice as a countable in this way nowadays.
If you are having trouble with uncountable words, it's often helpful to substitute an uncountable word you're more comfortable with, to see how it fits:
"The websites and helplines below can offer sugar and water ..."
It no longer makes semantic sense, but you can see how the grammar still works.