I really can't see how a man with the reputation of Vidal Sassoon would lie about it and try and pass another model off as Nancy Kwan.
I knew I wasn't imaging things about the programme I saw. .Here he his in his own words and pictures about her visiting his salon and cutting her hair.
https://www.bbc.co.uk/programmes/p00m73vb
Warner brothers wanted her to star in film called The Wild Affair in 1963 and contacted Sassoon to cut her hair and you can see her in that film with that haircut. Clips on Youtube.