I work in a training store that has had quite a few grad interns pass through it. Their start dates seem to vary. Some start right away and some don't start until after passing their boards. The pharmacy supervisor seemed flexible with them but I'm sure that is dependent upon the district.
One thing to be aware of is the amount of training you receive can be directly related to how quickly you take your boards. This past year two of the grad interns put their boards off until the end of the summer and trained in the store for a couple months. They learned the rxconnect system, completed required training modules, a staff pharmacist was always on duty to shadow their verification training, etc... Another grad intern passed their boards right away before setting foot into the pharmacy and as soon as the test results came back they were sent out to work in a store by themselves. They hardly had any training. Once you have your license CVS isn't concerned with the amount of training you have, they will just put you to work.