For Susannah Fiennes, travel is about looking harder. it is about being ravished by the moving geometry of bicycles in Shanghai, the “scaffolding and floaty bits” of tango dancers locked together in a club in Buenos Aires, the serenity of a still, blue-robed figure in the hazy warmth of Oman.

Sitting under electric lights in her studio in south London on a wet autumn day, these exotic locations seem to be from another world. On the easel and pinned to the wall are a couple of nearly finished oils that focus on light flooding through the archways of a mosque in Uzbekistan. The heat is palpable.

Yet, despite the extraordinary sweep of her journeys in the past few years – China in 1993 on a BP Travel Award and then to Oman, Hong Kong and South America on the invitation of the Prince of Wales as the official tour artist – Fiennes is not impressed with how opulently foreign places are. It is the quality of her response, not the width of experience that matters. “To me, Clapham High Street is a feast, really almost too much to take in.”

One of her early projects came out of a trip to Blackpool. “Ten days produced about two years’ worth of work. I loved the horizontals made by the sea and the promenade and the tramlines and seeing how a constant frieze, a classical frieze of people, would move across with coloured bags and push-chairs…”

The watercolours she produced do give a kaleidoscopic impression of crowds on the move and scattered about the beach against the big wheel and the tower in the background. In the Blackpool Tower Ballroom, she painted pictures looking down at the dance floor and tight clusters of couples in patterns as joyful as their Argentinian counterparts.

Later, Fiennes was employed as an artist in residence by a bank in New York, where she enjoyed the change of scale the city offered. I spent my time there looking out of the windows from a room on the 21st floor of a block on Madison Avenue and painting what I saw along the great canyons between the buildings. I’d often be the last one left in the building at night – there was something magical about seeing flickering, ephemeral lights come on in the skyscrapers.”

There is something fearless and determined in Fiennes’s approach to her travels as an artist. She is a second cousin, after all, to the explorer Sir Ranulph Fiennes and further back in the family history is Celia Fiennes who was the daughter of a colonel in the parliamentary army during the Civil War and who travelled the length of the land on horseback. Her journal Through England on a Side Saddle, later became a bestselling travel book.

Susannah Fiennes was introduced and recommended to Prince Charles by her friend Emma Sargeant, who had formerly accompanied him on tours abroad as the “royal artist”. It is not difficult to see why having been present with the collection of her slides.

Prince Charles invited Fiennes to Oman for the visit to mark the 25th year of its ruler, Sultan Qaboos bin Said.

As a painter, she is a traditionalist. at the Slade School, where she trained in the early Eighties, she was one of just 20 students determined to learn the language and grammar of painting.

While the other 180 young Brit Pack prototypes were working on their video installations, Fiennes and her set (dubbed “anachronistic, a joke” by their fellows) sat quietly day after day in the life rooms “analysing the changing planes that made up the model’s knee and mixing the equivalent in paint.”

For her, painting was to be a humbling and hard-won discipline of training the eye to see sensations of light shades and colour values before, as she puts it, “weaving the pieces together, just as music combines notes to make melody.”

The royal visit to Oman lasted four days. Not much time in which to absorb impressions of the country, while trying to keep up with the furious pace of the royal itinerary. She was out in the markets in Muscat sketching and painting when she could: a folding stool under one arm, her paints in a chopping basket or plastic bag.

The studies for what I think is the most glorious painting in the forthcoming exhibition of work from the Oman visit, the blue-robed figure standing like a still column in the heat, were done at the side of an airfield when Prince Charles and the rest of the entourage were watching the Red Arrows in a flying display.

“You have to be quite resourceful,” says Fiennes. “I’ve learnt to paint standing up with paints balanced in the crook of my arm. Sometimes, there have been usefully placed bodyguards to hold the water for my brushes. But even at the end of the trip during the official farewell and exchange of presents at the sultan’s palace in Salala. I was at work. I got the Minister of Water Resources to pose for me.”

There is no formal brief for the artist on a royal visit. Fiennes says she rather wishes Prince Charles would say more about what he wants. As it is, she shows him some of the work on the flight home and he can choose the pieces he likes from that selection and from the later work that will be done from memory in the studio. She will work for him on paintings of the tango from this year’s trip to Argentina.

The only real difficulty for an artist enjoying this royal patronage seems to be finding a space to do the work and not being left behind. “Generally, the equerries tell me how many minutes I’ve got,” she says. “But I’ve often found myself racing after the motorcade waving a wet painting in the air.”

In Hong Kong, where Fiennes was with Prince Charles for the handover and where she painted some wonderfully informal studies of Chris Patten and the British contingent aboard the royal yacht, she was taken from the ship to Kowloon by a couple of officers and left there alone in the streets with her stool and bag. The work went well, but there was an uphill struggle through super-tight security to get back on the ship.

What is interesting about the choice of Fiennes as a tour artist is that she is not in the obvious sense, interested in the significance of the events she witnesses. Her paintings are free of topical references and there are certainly no flags or generals or glimpses of royal encounters – for her, an official line-up becomes compelling only when the rain turns the scene into a shiny landscape of wet umbrellas. She is not interested in reporting back in information, social comment or, as she admits, in history. Her painting is about painting itself, about honouring and developing her art.

This exhibition may be the last we will see in Britain for a while. Fiennes is off to work in America where she has commissions from clients who are prepared to pay serious money for portraits. She has already spent time in Texas and Colorado this year and thinks it is time to establish a new base. I hope we do see more of her work – in a world gorged on camera images and the illusion of reality they present, it is good to know a traveller still prepared to look.