I lived in London, in the 70's. I had my first daughter, and as we lived in a one bed private rental, the council gave us a lovely 3 bed maisonette within 3 months.
I lived there, I had a second child, and the council wrote saying that the policy was now to offer families, with young children, a property with a garden. They made sure there was no rent arrears, and that we had kept the place nice, and offered us a 3 bed house on a private road.
We stayed there 28 years, until we divorced, and I now live in a flat, in a nice area, with current DH.
My ex was rehoused in a smaller property.
The small block I live in is for over 40's, with no resident children.
It's a HA property, in conjunction with a private developer. Something built for the Millennium. The rents are higher than usual (but lower than a private rental would be), but it's got higher spec than usual.
We were credit checked and referenced before the offer, but it's a nice place, and I won't be moving again.
I found it easy in the 70's, but perhaps it depended on where you lived. Our borough were very good.