Due diligence checks on the main applicant and their dependants older than 16 years of age can take 3 to 6 months. It depends on how full and detailed the information you provide about yourself, your family and sources of income.
Grenada government is keen on attracting as much foreign investments as possible; they will ask you to provide additional information unless you are subject to refusal straight away. As your interaction will be through an agent, this can take longer.
It is therefore very important to take extra care when retaining an intermediary that will provide legal support of your application for second citizenship by investment. They will make sure you have submitted a full bundle of documents which eliminates additional requests for information.