Family Allowance was most definitely paid only for second and subsequent children.
http://www.cpag.org.uk/MakeChildBenefitCount/C hildBenefit_4.htm
Prior to, and during, the Family Allowance period there had existed Child Tax Allowances. This was not a payment but an amendment to a father's tax-free allowance. They were first introduced in 1798, were available for the first child as well as subsequent children, but were abolished in 1805. They were then reintroduced in 1909.
To quote Barbara Castle, then Secretary of State for Social Services (of the Labour Government), during the Second Reading Debate on the Child Benefit Bill in May 1975:
"
It gives me the greatest pleasure to introduce the Bill, which I am sure will be accepted on all sides of the House. It achieves a long overdue merger between child tax allowances and family allowances into a new universal, non-means tested, tax-free cash benefit for all children, including the first, payable to the mother."
As an addendum to my first answer, in 1976 Child Interim Benefit was introduced for lone parents, payable for the first child.