He will need those ambassadorial qualities to the full over the next two months. Jeffries finds himself stuck on the City treadmill, ahead of a stock market flotation later this summer in plans announced last week.
His next task, on thoroughly unfamiliar territory, is to introduce the near-60-year-old firm and his colleagues to potential investors.
It should be third time lucky. A first flotation attempt, in 1990, was grounded after 35 of Atkins' employees, working as road engineers, were stranded in Kuwait, following Saddam Hussein's invasion.
"It was," says Jeffries, "pointless to even consider a flotation, when the lives of our employees were at stake."
Another, in 1994, came to nought, after market conditions proved unsuitable.
Despite his enjoyment of the job in hand, he admits to some mixed feelings. The notion of repeating the same message dozens of times does not thrill him. The usual caravan that accompanies such an exercise, however - merchant bank Schroders, alongside Cazenove as the broker, and City analysts, institutions and journalists - should find little to worry them.
Over the past five years since Saddam threw a spanner in Atkins' works profits have more than doubled, to pounds 19m from pounds 8.8m in 1991; fees last year came in at pounds 196m, up from pounds 154m the previous year.
Jeffries took over as chief executive last year, after his predecessor Colin Haylock left, following a boardroom split over the flotation and future strategy.
City investors will also be intrigued by one other unusual feature: employees own the bulk of the shares, with a third of the staff accounting for almost 60 per cent of the shares. The sellers are the original family owners, with their remaining 20 per cent stake, and the company's pension fund, which controls the other 20 per cent. Employee ownership has proved a powerful factor in the group's success, says Jeffries.
It is a long way from his birthplace of Jarrow. An unmistakable trace of Geordie in his voice - which gains full force whenever he returns to his native Tyneside - belies a cosmopolitan upbringing. At the age of nine, he was whisked off to Trinidad, in the Caribbean, where his father was chairman of a group of paint companies.
After school, and A-levels in maths and physics, he returned to London and the then Northern Polytechnic, where he enrolled on the arduous course to qualify as an architect. "My father, and his father before him, were engineers," he says. But he opted for architecture, as a bridge between the practical bent of his family, and his own strong interest in the arts. It took him nine years, including one day off a week, after he joined John Laing in the early 1960s for his first job.
He paints watercolours, which he sells through local galleries and fairs near his home in Kingswood, Surrey, an easy drive from the firm's headquarters in Epsom - and also plays the piano. His other interest is clocks, which he repairs himself.
Even so, there is little of the architect-aesthete about him. And he picked up from his father a strong engineering ethic, which has stood him in good stead at WS Atkins. He says of his own numeracy: "It's the only reason I think I have survived at WS Atkins." An imposing presence may also help, and a love of the variety management brings.
Founded in 1938 by Sir William Atkins - whose presence still casts a shadow over the company - it was his entrepreneurial style that set the tone for the company.
Jeffries' own instincts in this vein, however, were not satisfied in the early 1970s, when other jobs included a brief spell at London Transport. "I got in and out of there very quickly," he recalls.
Now 51, Jeffries joined Atkins' architectural firm in 1975, and by 1978 was head of the practice, with 150 staff beneath him. But it was as a business-getter that he made his name at the firm.
In the late 1980s he was transferred full time to business development - which involved lots of travel with the firm's worldwide spread - and in 1992, he was appointed to the board as marketing and business development director. Within just three years he was appointed chief executive - a meteoric rise, albeit interspersed with boardroom ructions.
As a business-getter, Jeffries was in the right place at the right time when Haylock departed, and graciously lays much of Atkins' recent success at the door of his predecessor.
Much of that growth has been organic, but Jeffries is keen to build by acquisition. In January, Atkins bought Faithful & Gould, a leading quantity surveyor, for pounds 21m and the flotation will make further such expansion easier.
Unlike many professions, much of Atkins' success derives from financial discipline, evident from its ability to ride out the recessions of the late 1980s, and early 1990s. For example, the company takes a vigorous line on accounting for project income. No profits are taken until a project is at least 50 per cent completed - a temptation which architectural firms especially have been susceptible to.
A comfortable cash cushion of pounds 85m suggests the discipline has paid off. It also gives Atkins the muscle to invest in projects itself, including the Government's Private Finance Initiative. Jeffries believes the bad press the PFI has had is unjustified. "We've applied our brains to it, we believe we can make money from it, and we are the preferred bidder on several projects." The firm has won its first contract - to build a new prison at Bridgend, in Wales, with Securicor and Costain.
The Thatcher years were good for WS Atkins. The disposal of Britain's major state assets proved an unexpected boon. Many companies that relied on public sector work for their bread and butter were squeezed. WS Atkins has flourished, however. Its network of government contacts, from Whitehall through to the big nationalised industries: rail, steel, electricity and nuclear, to the Ministry of Defence, and the Property Service Agency, have paid off in the new era.
Further afield, Jeffries is especially proud of the Chicago Beach project in Dubai, a man-made island which is the focal point of a $700m (pounds 455m) luxury tourist resort - although its design would be enough to make any self-respecting post-modernist shudder. Never mind: with WS Atkins enrolled as architect, engineering consultant and construction manager, it should prove a profitable venture.
Jeffries says the public sector still accounts for over half of revenues - but the emphasis is increasingly on private sector business, especially in the area of support services. Many are high-profile and politically sensitive. Nuclear waste disposal - the company manages the Dounreay nuclear site - the Channel Tunnel, and the Saudi Arabian Al Yamamah defence project suggest the company needs skills other than just engineering expertise.
Jeffries shows a diplomatic reticence answering questions on such topics. Of the difference between doing business in the private sector and with government, he says: "One has to understand the differing types of accountability within government, which is quite different from the private sector. And, as ever, you have to understand your clients' business. You must also understand how decisions are made, but I don't think it is possible to influence those decisions."
A tactful reference, perhaps, to former transport minister Archie Hamilton, a paid consultant to the firm, and whose activities were highlighted in the wake of the Nolan Report into MPs' private interests. But nor should that deter the City.