I suppose the phrase that would substitute "Heading to The City" or "Going up to town" over here would be "Heading North". We all know what that means.
Out of curiosity, where is "over here" for you?
I've lived in many different parts of England, and rarely have people referred to the nearest big settlement as 'The City', though to be fair most of the places I've lived
were the nearest big settlement. When people talk of "going to the town", they mean the town centre (interestingly, this is the case even in cities, probably a relic of when they were lots of small towns that have since been swallowed up). The only place I've heard people talk of 'the city' was when I lived in the suburbs of London, though most often it was known as 'Central London', or just 'London'. The very centre, the historical "City of London" is indeed known as "The City", but not really colloquially, only among twatty businessmen.
All that aside, however, I agree with Interlude's assessment.