There are few things to make one feel more like a hapless Englishman abroad than the realisation one has the wrong currency.

Worse still is when the realisation you’ve been given Norwegian Krona at the airport occurs in a crowded Danish restaurant. It’s a bit like saying “Norway, Denmark, Finland – they’re all the same aren’t they? Which one are we in again?”

Mercifully, the people of Copenhagen couldn’t be more accommodating and the waiter couldn’t have been more delicate in explaining the situation to his blushing guests.

I’m told it isn’t simply the Christmas spirit that keeps the Danes so wonderfully neighbourly; everyone I know who’s visited the country, regardless of season, has expressed similar sentiments. And yet there is a detectable festive crackle in the air in the run-up to Christmas in Copenhagen. It’s a city that makes the most of the season, from the spiced glog (strong mulled wine) available on every corner to the feast of illuminations that is the Tivoli Garden – one of the oldest amusement parks in the world.

For me, though, these were but the dusting on the chocolate yule log in a city whose breathtaking wintry beauty has the kind of perfection you’d expect from a postcard or oil painting.

The Danes have a word that finds no easy translation in English – “hygge”. It’s a more sociable, atmospheric cousin of our “cosiness” and evokes – among many other things – that feeling of gathering in small rooms around a fire, drinking piping hot booze and unashamedly indulging in the bonhomie of being crowded together around the hearth. It sounds too slushy to be true, and yet we saw it any number of times in the restaurants, bars and cafés around the city.

The hygge dial had been cranked right up at our first restaurant, the Café Sorgenfri, where groups of men gathered in large groups for herring and Schnapps. Even our excruciating Norwegian currency episode was unable to disrupt the atmosphere in the small, low-ceilinged room.

On the highest reaches of the hygge scale is Nyhaven, the city’s maritime quarter. The timbered buildings (one of which was home to Copenhagen’s most famous son, Hans Christian Andersen) house restaurants and cafés, and play host to one of the city’s many Christmas markets. The festive food of choice on offer couldn’t be more warming – roast duck stuffed with prunes, baked apples and spiced red cabbage.

Nyhaven is also the point of embarkation for the city’s regular canal trips. Reasonably priced, the trips visit the places that would be time-consuming to go to on foot, including the not-terribly-impressive statue of the Little Mermaid, thought to be one of the most photographed statues in the world.

Of far more interest is the strikingly modern architecture that lines the waterside, centuries and a thousand aesthetic miles removed from the narrow, pastel townhouses of the old Nyhaven harbour. This, after all, is a city known for its elegant sense of form and function. The globe-like glass front of the Copenhagen Opera House (one of the most expensive ever built at an eye-watering 500 million US dollars) gazes out over the water, along with the complex outline of the National Theatre’s new playhouse.

It’s this elegance in design that reins in the tackier tendencies of the festive season here; throughout the city, Christmas hung in the air but was never sickly. Our hotel, The Square, which overlooks the town square with its clock tower and mammoth Christmas tree, was all clean lines and sophisticated comfort. But there was a very conspicuous exception to all this urbanity – the unashamed tinsel-fest that is the Tivoli Gardens. A kind of proto Disney World the rest of the year, Tivoli becomes something rather incredible during Christmas.

If you’ve ever seen Tim Burton’s The Nightmare Before Christmas, you’ll remember the moment when Halloweentown resident Jack Skellington tumbles headfirst into Christmas Land and is confronted by a snow-covered, treacly new world. That’s what it’s like here. The glogg still flows, but everywhere else it’s all about a childlike wonder, with the park’s sprawling pavilions and sculptures festooned with endless lights.

Elsewhere, there are costumed japesters and an illuminated lake, and it genuinely is one of those things you have to give in to and enjoy. But a word of caution: eat at one of the restaurants outside the Tivoli gates rather than the overpriced eateries within.

Most of the Christmas markets can be reached in a day’s walk around this compact city. Don’t expect hundreds of stalls, though; the markets comprise no more than a dozen stalls each time. It’s actually much nicer to stumble across them than to be overwhelmed in one place.

It’s a wonderfully compact place to stroll around, yet also relatively spacious and simple to navigate. Most refreshing was the contrast between Brighton’s treacherously icy winter streets and the clear pedestrian walkways of a city in the grip of similar freezing temperatures.

A jaunt through the city’s laidback, café-lined university quarter gives way to the Rundtaarn (round tower), an architectural wonder of the 17th century. The oldest functioning observatory in Europe, the tower’s spiralling interior was once home-from-home to Tycho Brahe, the visionary Danish astronomer whose observations at the Rundtaarn constitute some of science’s most important discoveries.

Four centuries later and the Rundtaarn offers a panaromic view of a city whose skyline has been fiercely guarded against the concrete towerblocks deemed necessary in more densely populated conurbations.

The result in winter is a gorgeous patchwork of medieval snow-covered houses, modern roofs and stretches of park – a perfect 360-degree view of this giddying, romantic place.

* Fly to Copenhagen from Gatwick in less than two hours with low-cost airline Norwegian. Prices start at £29 one-way, visit www.norwegian.com

* For more information on Copenhagen and Denmark, visit www.visitdenmark.com, www.visitcopenhagen.com and www.thesquarecopenhagen.com