Mr Vineall retired as head of management development and training for Unilever four years ago. Since then, he has been spending one or two days a week as chairman of the Tavistock and Portman NHS Trust, doing some personnel consultancy work and spending his precious free time walking, gardening and improving his bridge.
Now what would persuade any sensible person of 64 to abandon that well-balanced life for what must count as one of the most thankless unpaid tasks in the public sector? With schoolteachers at all levels demanding more money and better conditions on the one hand and Kenneth Clarke, the Chancellor, threatening review bodies with the rack if they dare recommend anything over 3 per cent, what is the hope of making anybody happy?
But Mr Vineall has a strong streak of public service in him. He can justifiably feel that his experience running pay policy for Unilever's 300,000 employees worldwide has equipped him to tackle the complexities of schoolteachers' pay. And he has done this sort of thing before, first as a member of the Review Body on Doctors' and Dentists' Remuneration, then as a member of last year's Bett Review on pay and career structure in the armed forces.
What is more, he thinks it is worthwhile. In both the health service and the armed forces, he was struck by the almost universal support for the existence of the review body - the feeling, as he puts it, that "life was better with review bodies than without".
The feeling seems to be justified. The Department for Education and Employment's evidence to the School Teachers' Review Body says that public sector workers with review bodies have done much better than others over the past year.
Mr Vineall is unwilling to confirm or deny that. Understandably, any question on the ticklish subject of public sector - especially teachers'- pay is courteously fended off.
Is he worried by teacher shortages and early retirement? Is he prepared to tackle conditions of service? "At this stage, I'm probably at the peak of wanting to learn more before being able to answer that question," is one typical reply.
But perhaps he could make a general statement about his approach to pay, for instance at Unilever? Does he, for instance, believe in paying well at the top to motivate those lower down?
"It takes all sorts to run a business," he replies, "and at different times the pressures fall at different points." Hard to argue with that.
The son of a chemist and former schoolteacher, Tony Vineall was educated at Leeds grammar school and New College Oxford, where he read philosophy, politics and economics. Both of his sons went to state primary schools and then to the Royal Grammar School in Guildford, which was maintained when the elder son started there but independent by the time the younger joined him. ("I am non-doctrinaire on these issues," he murmurs). He is still a governor of the school.
Mr Vineall is currently wading through the written evidence before he and his team start taking oral evidence in October. But the impression he has made during first contacts with the teacher unions suggest a chairman more open than his predecessor, John Gardiner.
"He's much more interested in people and making things work than John Gardiner was," said one union official. "John Gardiner did a fair-minded job within narrow limits. We hope Tony Vineall will take a broader view and will tackle the conditions of service issues that have been left untouched."
Julia Cleverdon, formerly of the Industrial Society, (where Mr Vineall chaired the executive committee), is more fulsome. "He's a very decent, honest and high-integrity man," she says. "Personally, he's quite humble."
That humility must explain his absence from Who's Who. His new job should see him finally join the pages of the Great and the Good.