What follows is the product of thoughts which have been passing through my head. Without anything more than the written evidence of Pitt paying for Elizabeth, and her younger sister Louisa Jane's, education, clothing, and boarding for at least the period of 1797 to 1801 (and in Elizabeth's case - until Pitt's death in 1806), this must be viewed as historical speculation.
If William was Elizabeth's father (she was born on August 3, 1785), then she would have been conceived in the autumn of 1784. Pitt had been Prime Minister (First Lord of the Treasury in those days) for a year by the time she was conceived. Elizabeth's mother was Jane Pryce Williams - a married woman. Her acknowledged father was Edward Williams.
Under English law at the end of the 18th century, children were considered to be the property of the mother's husband, and whether the husband was the father or not, the children would definitely have shared the husband's surname. Thus, any illegitimate child borne to a married woman could easily be passed off as her husband's, and Pitt would have been absolved of any responsibility for the child - and Jane protected from the stigma of bearing a bastard. Pitt would have had good reason to be relieved that his paternity of Elizabeth did not have to be acknowledged; he could preserve his carefully cultivated image of being a virtuous minister.
Jane died in early 1797, and quite ironically, it is from this exact period which Pitt took both of her daughters into his house and began paying for every aspect of their upkeep and education. He was a generous man, but there is absolutely no written record still in existence of him extending this same provision to any other children.