GLASGOW — Pushing purposefully through the crowd, Scotland’s first minister, Nicola Sturgeon, stops abruptly, her path blocked by a well-wisher cradling a photogenic 5-month-old. Without hesitation , Ms. Sturgeon gathers the baby smoothly in her arms and slowly plants a kiss on his forehead, as the cameras click in unison.

Seen by many as Britain’s most effective party leader, Ms. Sturgeon is not even running in the Dec. 12 general election because she sits in the Scottish Parliament in Edinburgh, not the British one at Westminster.

But as leader of the buoyant pro-independence and anti-Brexit Scottish National Party, she is the face of its campaign for Scotland’s 59 parliamentary seats in Westminster. The success of that campaign could determine whether Britain leaves the European Union in January — and, if it does, whether the United Kingdom survives the rupture.

At a recent and well orchestrated visit to a charity in a gritty part of Glasgow, Ms. Sturgeon was everywhere, helping out at a numeracy class, performing a gym workout and, in the kitchen, ladling out bowls of thick lentil soup.