Tucking into a steak - her schedule for the rest of the day is so punishing that she may not have time to eat again - she outlines a plan to invite hereditary peers to select 40 of their number who would remain in an interim chamber. This, she reminds me, is what the Scottish peers did at the time of the Act of Union. It sounds fair to the opposition for someone engaged in the hurly-burly of party politics, and suddenly Lady Williams laughs mischievously, admitting it isn't official Lib Dem policy. "It's a bit radical," she concedes. She gives the impression that she is enjoying this election enormously.
Earlier in the day, at a briefing in Westminster on women's issues, she makes serious points about the adversarial nature of the contest so far. "We get the impression out on the streets that a great many women voters are turned off by this." Flanked by the Lib Dems' campaign co-ordinator and fellow member of the House of Lords, Richard Holme, and two of the party's most personable women candidates, Lady Williams talks about putting a penny on the standard rate of income tax to fund pre-school places for three- to five-year-olds, reduce class sizes to 30 and provide more textbooks and computer equipment. She restates the party's intention to raise the tax rate to 50 per cent for people earning more than pounds 100,000, which would allow the Lib Dems to take 450,000 poorly paid workers out of the tax net altogether - many of them, Williams says, women in part-time work with family responsibilities.
It is a low-key occasion - sparsely attended, like any election meeting called specifically to address women's issues - but rich in irony. Here is Shirley Williams, the former Labour cabinet minister regarded for many years as a traitor for breaking away in 1981 to found the SDP, staking out territory that places her well to the left of Tony Blair's New Labour on a host of issues, including the explosive one of tax. It has been an unpredictable political odyssey, as Lady Williams is aware, yet her forceful appearances on Question Time and Newsnight in recent weeks suggest that, at 66, she is enjoying a political renaissance. During the briefing and on the journey to Liverpool, I suspect that this is because she talks a language - about social justice, about confronting the problem of poverty, about this country's shameful treatment of asylum-seekers - which both main parties run a mile from.
It may also be because there is a longing, among women voters in particular, for confident female voices which do not just articulate their concerns but show that women can hold their own among the Johns, Michaels and Tonies. John Major's first cabinet in 1990 was shockingly empty of female ministers while New Labour is becoming famous for its Silent Women. Lady Williams recalls a presentation on Labour's economic policy by Gordon Brown and Margaret Beckett when "Margaret couldn't get a word in edgeways. It gives people the wrong message."
Since 1988, Lady Williams has held the post of Professor of Elective Politics at the John F Kennedy School of Government at Harvard University. (Her second husband, Professor Richard Neustadt, is American and the couple divide their time between the United States and England.) With an academic's grasp of polls and statistics, she points out that women are not only a majority of the British population - 52 per cent - but that their votes increasingly determine the outcome of elections. Last November, there was a startling 18 per cent gender lead among women voters for Bill Clinton, while men gave a small lead (1 per cent) to his Republican rival, Bob Dole.
In Britain, though, Tony Blair is failing to win women's votes. While New Labour is ahead of the Conservatives, its lead is on average six points lower among women than men. The party's image is still male-dominated and Lady Williams's return to a high-profile role in the Lib Dems' election campaign emphasises this weakness of her old party's front bench. When I suggest that voters are in danger of electing Tony Blair for the wrong reasons, as Tory MPs did John Major in 1990, she agrees. Her fear is that we are living through a period of soap opera politics, in which faces change - unpopular characters are killed off - but not the plot. The campaign, she thinks, has not generated anything like the excitement that characterised the 1964 election - another moment when Labour, this time under Harold Wilson, looked like taking over after a period of Conservative rule. "One of the reasons I was in the Labour Party for so many years was its commitment to social justice," she says. "I'm not saying it's not there any more but they are not prepared to spell out the consequences of it. They are not prepared to say to comfortable middle Britain that it has to make sacrifices. Without rubbishing Labour, we owe more to the people who are losing in our society than just words."
This is, of course, the view of a politician who is campaigning for another party - and one that has no prospect of forming the government. "The polls are narrowing," she says, "but I don't think there's much likelihood of Labour losing." A Labour landslide would be a disaster, she believes, dashing the Lib Dems' hopes of having a serious influence - by which she clearly means a radicalising influence - on Tony Blair's administration. "I know there's a lot of Labour MPs who would welcome it," she adds, while refusing to name them.
She has finished lunch and, reluctant to waste a moment of the election campaign, she takes out a sheaf of papers and prepares to work on them for the rest of the journey to Liverpool. She gives her husband a rundown of the next couple of days' meetings in Merseyside and North Wales and he nods, enjoying this firsthand experience of British politics. As I leave them, Shirley Williams grins at him and says: "We're just a couple of old crusaders, aren't we darling?"