It depends entirely on the name of the country and its syntactic construction.
✔ The UAE.
→ ✔ The United Arab Emirates.
✔ The U.S.
→ ✔ The United States.
Both of those are acronyms for longer names that work with an article.
✘ The England.
✘ The Canada.
✘ The America.
None of those are names that work with an article. Also note that both the United States and America describe the same country, but the name itself is what determines the acceptability of an article.
In short, if the country has a common-noun equivalent, then putting an article in front of it will be at least idiomatic, if not actually grammatical; if it has no common-noun equivalent, then it won't be.
When you use a possessive, you simply add 's to the noun. It's no different here. So if the proper noun uses an article, then so does its possessive form; if the possessive form doesn't use an article, then its possessive form doesn't.
On the other hand, if an article is optional with the proper noun (it's fine both with and without it), then exactly the same is true of its possessive form.
At the risk of abusing a frequent example of an uncommon situation, take the first name of the president of the United States:
✔ Donald's speech.
✔ The Donald's speech.
Since that is a specific example of a proper noun where both omitting the article and using the article are equally acceptable, it's also acceptable to omit or use the article in its possessive form. (This was the only example of such a proper name I could think of; I don't know if there's a country that follows the same optional pattern.)
But that's an unusual example. Normally proper nouns either do or don't take an article—and their possessive forms match that.