The registration process itself will be started usually immediately, since we send it in real time to the responsible registry. In most cases the domain will be officially registered within a few minutes. For some TLDs such as .io (Indian Ocean) or .cx (Christmas Island) it may take a few days for your domain to be registered, because the responsible registries are not yet fully automated. After the completion of the registration process, it will take a few minutes (up to 2 days, depending on the TLD), until your domain is globally available/reachable.
More details can be found on the detail page of the respective TLD. You can find it via the overview of all available top-level domains.