There is no single answer here, because the grace period is set by each country and often depends on your residency status rather than just time passed. A very common pattern is that tourists can drive on a valid home license, with or without an IDP, for somewhere between three and twelve months, but the clock changes the moment you become a resident. Many places require new residents to convert to a local license within a fixed window, sometimes with a driving test, sometimes as a straight exchange depending on reciprocity agreements between countries.
And yes, nationality matters a great deal. License exchange treaties are bilateral, so a license from one country might be exchangeable without any test while another nationality faces the full written and practical exams. The language your license is printed in also affects whether an IDP or certified translation is required alongside it.
The definitive source is the destination country's transport or licensing authority, or your embassy's driving guidance page. Rental desks and forum posts get this wrong constantly, so verify against the official rule for your specific license before assuming you are covered.