Capitalization (especially of titles) is a tricky business in English. There is no hard-and-fast rule about capitalization. In fact, many of the major style guides disagree on exactly what should or shouldn't be capitalized. However, I'll give some general guidelines and let you make up your own mind:
Titles in direct address
In all major style guides, when a title is used as a name in direct address of that individual, it is capitalized. For example: "That was very kind of you Mom." "I love you Dad." "Stop being so good to me, Master." "So, I have to have surgery, Doctor?"
Titles used in direct reference
When referring to a specific individual with their title, the title should be capitalized. For example: "I was excited when President Barrack Obama walked in the room." "My favorite comic book character is Mr. Tony Stark." "It took a while, but we finally got in to see Doctor John Smith."
However, (somewhat less commonly) some prefer that occupational titles not be capitalized, unless abbreviated. For example, "Dr. John Smith" and "doctor John Smith".
Titles used in indirect reference
Most style guides prefer titles used in reference to be lowercase. For example, "I love my mother." "The president gave a great speech." "My doctor did a great job with my surgery."
However, some titles are exempted from this. God (when referring to the Judeo-Christian god) is always capitalized. The King and the Queen are often capitalized. (For example, "The Queen has quite a way with words.") When referring to a specific office, there is disagreement about what should or should not be capitalized. Chicago and Wikipedia prefer "the president of the United States", while NY Times prefers "the President of the United States". However, everyone prefers "King of France" to "king of France".
Why is Uzi capitalized? It comes from a name, and people haven't frequently used it in lowercase in publication.
First, the name is derived from a person's name. These usually retain their capitalization. For example, we have:
- Tommy gun, or the Thompson submachine gun, for inventor John T. Thompson (Wikipedia)
- Molly or Molotov cocktail, in mockery of Soviet foreign minister Vyacheslav Molotov (Wikipedia)
- The Luger, for inventor Georg Luger (Wikipedia)
Even though they come in multiple models and forms, the type of weapon retains the capitalization of the name, even if it's not also the name of the manufacturer.
Furthermore, Uzi hasn't generalized to the point of being lowercase, like some other products. For example, we have:
- diesel, in diesel fuel and diesel engines, for inventor Rudolf Diesel (Wikipedia)
- leotard, for performer Jules Léotard (Wikipedia)
- voltaic pile, for physicist Alessandro Volta (Wikipedia)
Popularity makes lowercase more likely, but shifts in usage are arbitrary and vary between individual words. For instance, Ferris wheels are well-known, but the name retains the capitalization.
With Uzi, too, the capitalization remains: in a News on the Web Corpus search, among 2226 results for "uzi," virtually all results are "Uzi." Only 31 of the first thousand results were lowercase. So it will be a while before one can build an argument for "Uzi" being a valid Scrabble word.
Best Answer
Technically, 'constitution', (like 'president') is not a proper noun so it shouldn't be capitalised. But American veneration for their political system's institutions is such that these terms have taken on the status of proper nouns. And the decision to capitalise (or not) such terms is codified in media style guides. The (UK) Guardian takes a somewhat iconoclastic view in such things (even for UK institutions), often choosing not to capitalise these terms, and not even 'queen' unless it's referring formally to Queen Elizabeth II.
In answer to your question, then, as a non-US person, I would recommend not capitalising 'constitution' at all, ever, unless it's the first word in a sentence!