I wonder if Eliza was illegitimate? From IGI:
ELIZA VAUGHAN NIGHTINGALE Birth: 26 MAR 1810/ Christening: 20 JUL 1823 St Giles Cripplegate, London
Parents: HANNAH NIGHTINGALE
If she used Vaughan as her middlename (like JBL used Ballard in his reporting), may explain why the children of JBL and Eliza showed in their christenings on IGI with mother's name as Eliza Vaughan (which I read as surname but actually could well have been her middle name).
Monica