The length of time it takes to open up AME, load the comp, start dynamic link to open a background copy of AE to do the rendering, and start the render depend entirely on your available system resources. Right not I am using Photoscan to create a 3D model from 695 images, taking a break from doing some hand roto in After Effects on 8K Red Raw footage, and looking at this forum. If I were to send one of those 8K comps to the AME while Photoscan is running and I'm working on the web I would expect it to take five to ten minutes. If all I have open is a web browser and mail and my comp is HD it takes about 90 seconds for AME to wake up, load the comp, an be ready to start the render for the first time. Photoscan is using about 80% of my system resources and memory so there's nothing left. That may be the case with your project. Only Project details will let us know:
Safari, Photoscan, then AE and almost all my system resources are gone.