Although trademark registration can take anywhere from 4 to 27 months, depending on the country, luckily, you only need a pending status to join Amazon Brand Registry.
After your trademark application is filed, the intellectual property office will give you a serial number that you can submit to Amazon. It usually takes a couple of business days for the number to be recorded in the IPO's database and start working.
You can check the status by searching the database yourself since trademark registries are publicly available. Once you see your trademark there with a pending status, your serial number should be valid.
Keep in mind that if you are using the help of an attorney/law firm to prepare your application, you need to also account for the time it takes them to prepare it. The process can also be prolonged if your legal representative points out issues with your mark that will need solving for it to be registrable.