It would seem to me, from reading the responses, and from thinking further, that the history of the development of New York City played a big part in the notion of 'Downtown'. It is a word that generally-speaking sounds foreign to British ears, but which is employed by the travel industry in the UK for the benefit of tourists.
'Downtown', as I understand it, has two important senses. 'Downtown Minneapolis' is what in Britain we would call the 'City Centre of Minneapolis'.
Where Americans say they are 'going down town', we (especially Londoners) will say we are going 'up town'. In other British cities people may say 'I'm going into town', or 'I'm going in to the City Centre'. But usage across Britain is highly peculiar to local circumstances, and varies greatly from place to place. Also the whole question of what is a city and what is a town, and in what circumstance we refer to Birmingham as a city and in what circumstances we say 'going into town (meaning Birmingham centre) ' is highly nuanced and particular to places.
I would also be interested to learn exactly what is meant by the American term 'Central Business District'. In the case of London, I am never sure if this would mean the City of London, where the Bank of England, the Stock Exchange, and the whole panoply of financial institutions are housed, or the shopping area around Oxford Street.