They are not always interchangeable. In your question you gave two examples. "The only one" is something you will see written and hear spoken very often. You would never see or hear "the mere one". It is not a correct use of the word. "Merely" most often emphasises the lack of importance or relevance of a thing. Examples: "The only important thing he said came first, everything was merely additional information" "We know who will win, the rest are merely making up the numbers"

I do not think these words are interchangeable and I will give you examples below that support my statement. I'm sorry if I provide any inaccurate information as this is a topic that is hard to explain since I'm native. "I only like to eat apples" - This sentence implies that I will only eat pizza, but what would happen if I replaced "only" with merely? "I merely like to eat apples" - While this sentence implies that I barely like to eat apples and would rather eat something else. I think of it as "merely" points to hardly or barely. While "only" points to just that subject.

You can usually replace 'merely' with 'only', but not vice versa.