Nuki’s smart locks mount on the inside of the door over your existing lock and open over Bluetooth from your phone. Nuki’s privacy policy says an account is not mandatory to use its devices, and current models also support Matter, so a local smart home can control them without Nuki’s cloud.
Remote access and Nuki Web do go through Nuki’s servers, and the Nuki Box needs an account. Check that your door and cylinder are supported before you buy, because the fit depends on the lock you already have.