Noting this line in the OP ...
"... City Centre (equivalent to Downtown with capital D)"
note that "downtown" and "uptown" have no connection to city "center"
these two words are today adjectives that mean a certain type of style.
Downtown: an "artistic", "bohemian" style. for example, the "artistic", "bohemian" expensive part of a city center
Uptown: a "straightlaced", "formal" style. for example, the straightlaced expensive part of the city center
The epitome of this is of course in New York ... downtown is SoHo, Tribeca and so on (central / expensive / bohemian). Uptown is Park Avenue and so on (central / expensive / straightlaced).
(Both parts are fully "central", they are both the densest parts of the urban center. If you're in say Connecticut, and you're about to head to Manhattan, you wouldn't say "you're going uptown" nor would you say "you're going downtown". You'd say "I'm going in to the city" (or perhaps "I'm going to Manhattan"); someone may ask "where are you going in the city?" and you may answer "I'm going uptown" or "I'm going downtown".)
They're, today, just adjectives.
For example, https://www.youtube.com/watch?v=hCuMWrfXG4E
(Or indeed https://www.youtube.com/watch?v=OPf0YbXqDm0
A new coffee shop (anywhere - in Sydney) might be described as having an "uptown vibe" (stainless steel, etc) or a "downtown feel" (bohemian). Andy Warhol - downtown. Donald Trump - uptown.
This usage surely originates with NYC becoming a general cultural referent, at a guess in the 1970s? (The distinction between SoHo, Park Avenue, was in place by then.)