Under the retro body a modern MINI is a BMW, and it treats its battery the BMW way. A sensor on the negative terminal reports charge and battery age to the car. When a new battery goes in, it has to be registered with a diagnostic tool so the charging system starts afresh. Skip that step and the car keeps charging a new battery as if it were worn out, which shortens its life and can leave start-stop disabled.
Location varies. On recent Hatch and Convertible models the battery is at the back of the engine bay under a plastic cover near the windscreen. On recent Countryman and Clubman models it is commonly in the boot under the floor. First-generation Cooper S models also carried it in the boot. Most start-stop MINIs use AGM. Because MINIs are often second cars doing short trips, the battery spends much of its life half charged, and heat finishes the job.