From this Day Forward — Beryl Matthews

As if everyday life in London's slums wasn't hard enough for Jane Roberts and her three children, the day that ne'er-do-well husband Bert walks out on his family as rent day draws near is a particularly dark time. In the wake of his desertion, the family pull together to find their way forward, and Jane takes the steps she needs to get a divorce and free them all from Bert in the future.
With help from a neighbour who is not all she seems, a lucky break and fresh opportunities, the road ahead starts to look more promising. But will Bert wreck the new start the Roberts family have their sights set on?