Is it possible to recover from breast augmentation alone in the week I don't have my kids?

You were not specific about what you will need to be doing when you have your children. No, you are not healed from surgery, but you have recovered from the anesthesia. If you are no longer taking pain medication or muscle relaxers, you should be able to drive. You will not be able to lift children up to 6 weeks. You should be back to most activities by 6 weeks.

The short answer is 'yes' but keep in mind that recovery times will differ greatly depending on the surgeon you choose.

Recovery from breast augmentation can take four to six weeks. During your recovery, we recommend only doing low-stress activities for the first two weeks, such as short walks and regular household activities. At two-weeks post op, we recommend you begin some low impact cardio like exercising with an elliptical or stair master, or hiking if you prefer being outdoors.
After four weeks, you can begin to reintroduce resistance training with no chest exercise. For breast augmentation patients, we recommend no chest exercises until about eight to twelve weeks after surgery to ensure that the precise location of the implant does not shift in an unwanted direction.