Ms Baxendale is counsel for the inquiry, which since May has been examining the extent to which the Government connived at the export of defence equipment to Iraq, potentially breaching its own regulations. It is also examining allegations that the Government was prepared to see six industry executives go to jail for the alleged breaches of regulations rather than admit its own dishonesty. The exposure of the inner workings of British arms-dealing bureaucracy is causing severe embarrassment.
In the long sessions at 1 Palace Gate, Ms Baxendale performs an intimidating double-act with Lord Justice Scott. She is his hit-woman, leading the questioning of witnesses, the apparent mistress of more than 100,000 pieces of paperwork.
Victims of her deceptively soft inquisitorial style have so far included William Waldegrave, minister for open government, David Mellor, the disgraced former secretary of state for national heritage, and Tim Renton, one-time chief whip. She is likely to question John Major and Baroness Thatcher as the inquiry continues.
Ms Baxendale comes from south London, daughter of a man whose claim to fame, according to a family friend, was that he manufactured turkish delight. This made him enough money to send Presiley to St Mary's, Wantage, then a fee-paying Anglican convent school run by the nuns of the community of St Mary the Virgin. It was a small, friendly place, and Presiley was made a house leader. She went on to take A- levels at Westminster School, London, where even her name is only faintly remembered.
Westminster is known for turning out journalists and lawyers, and, sure enough, Presiley applied to St Anne's College, Oxford, to read law. St Anne's, which had become a university college in 1952, 18 years before she arrived, was one of five colleges that accepted women, and that, combined with the fact that it had a good reputation for law, meant she survived fierce competition to get in. Ruth Deech, the principal, taught Presiley and remembers her as cheerful, energetic and relaxed: she worked hard, but not too hard, and played tennis enthusiastically.
Members of her chambers and barrister friends extol Ms Baxendale's intellect. 'I have known her since before she was a student and she has always been extremely brilliant,' says Stanley Brodie QC. Sir Louis Blom-Cooper QC agrees: 'What strikes one immediately about her is her high degree of competence.' Such qualities apparently developed after university. 'She was a good student but I wouldn't have said brilliant,' admits Ms Deech.
After graduating with a 2:1, Ms Baxendale taught in south London before being called to the Bar in 1974 at 23. She went on to make her name in local government and public law cases, most notably acting as counsel for the tribunal in two child-abuse inquiries. One, in 1984, was into the death of Jasmine Beckford, who was beaten by her stepfather, and another was into that of Kimberley Carlile, a four-year-old who also died, in 1986, at the hands of her stepfather. During the first, Ms Baxendale was pregnant with her daughter Felicity; at the second, with her son Charlie.
Her husband, Richard FitzGerald, a tax barrister, is said to be charming and keen on carpentry. Sir Louis describes him as 'unlike her in many ways - austere and apparently content to be in her shadow'. He does not, unlike his wife, appear in Who's Who. They and their children, now aged eight and six, live in Regent's Park, central London, and spend weekends at their rambling country house in Wadhurst, East Sussex.
Those less happy to be in her shadow are her victims in court, who are subject to her water-torture questioning. They are learning the truth of a warning from Lord Lester QC, a senior member of her chambers at 2 Hare Court: 'She gives the impression of being a pushover. In fact, she is very strong personally. You could be deceived.' A former court opponent agrees: 'She's a quiet and very able operator - if it is necessary to turn the knife she's perfectly able to.'
Ms Baxendale's method is to work through evidence carefully, repeatedly asking the same questions of witnesses. This can drive them to distraction. Observers have noted her unsettling habit of giggling girlishly before going for the kill. On the stand on 22 September, Mr Waldegrave was left squirming. At one point he admitted: 'There is a misunderstanding here. An understandable misunderstanding.'
He added later: 'The only point I'm disagreeing with you (on) and I'm not sure I am disagreeing. . .' After another exchange, Ms Baxendale reassured him: 'I didn't think you meant what you were saying.'
Her politeness is legendary, but her views tend to the conservative; Sir Louis, who brought her on to the committee of ICSTIS, which monitors telephone services' morality, admits: 'She's pretty opposed to pornography. I don't think she ever wanted to ban lines, but she was quite keen to control them by coming to a decision that they went beyond what was acceptable.'
She did not always clamp down, however. Commenting on attempts to ban a game called Dial Dr Dark in 1989, Ms Baxendale remarked coolly: 'I find it extremely boring.
It makes no reference to chopping up one's mother or anything like that.'
She can be charming and down- to-earth, sucks her nails frantically, and is willing to apologise if she appears aggressive.
A letter in the Independent in May, which raised the perennial question no one appears able to answer - the derivation of her first name - elicited a chatty, hand- scrawled explanation from her. Unfortunately neither the recipient, Angela Turner, nor Mr Brodie, who has also learnt the secret, can remember what it was.
The mystery continues, but Ms Turner recalls: 'It was nothing to do with Elvis Presley. I have the distinct impression it was an old family name.'