Leave it with me
What it really means
Means "I will deal with it". From a reliable person it is a real promise. From someone busy or avoiding the task, the matter may quietly disappear — so follow up politely if nothing happens.
Context, tips & common mistakes
"Leave it with me" is a genuine acceptance of responsibility — the speaker has taken your problem onto their desk — but note what it deliberately omits: any deadline. It buys the speaker time and closes the conversation gracefully, and most of the time the thing does get done.
The etiquette of waiting: allow a reasonable interval (days for small things, a week-plus for big ones), then one polite check-in — "any news on the boiler?" — is fully expected and causes no offence; hovering daily does. If you need a date, extract it at handover time: "brilliant — roughly when should I check back?" When you say it yourself, treat it as a small contract: Brits remember who follows through on 'leave it with me', and quiet reliability is the local currency of trust.