You are experiencing paralysis of the left marginal mandibular nerve which controls the depressors or muscles that pull the lip down. The implant should have been under the periosteum directly over the bone which is a very safe plain to be in. If your surgeon used an incision under the chin than it's possible that the nerve could have gotten stretched causing a temporary neuropraxia or paralysis. It's unlikely that the nerve was cut.
So take heart your surgeon is probably right and the function will probably return in the next several weeks.
Your smile should go back to normal.
You certainly have weakness of your left lower lip, and this is one of the possible problems that can happen after Chin Implant surgery. Any surgeon that performs this operation regularly has seen this problem. It's not common, but the function of your lower lip will return provided the nerve wasn't cut or cauterized. This typically occurs from stretching the nerve while placing the implant.
I've seen this problem (fortunately not frequently), and it tends to normalize in less than 6 weeks. It could take up to 6 months, but that is not the norm.
If it's not better in a month, you can have a small amount of BOTOX injected in the right lower lip (5-10 units). This will provide symmetry if the problem persists for over a month.
I'm sorry you're going through this, but you should feel confident that in all likelihood, your smile will completely go back to normal.
Temporary nerve damage after chin augmentation.
1) You have a complication. Nerve or muscle damage in the left side of your mouth.
2) The good news is that this is usually temporary. In 4 months or so, you should be back to normal.
This is not uncommon after Chin augmentation and is more commonly associated with liposuction that can effect the nerve or muscles that work on the corner of the mouth. In most caaes this will improve by 3 weeks although it can take longer but general return to normal can typically be expected.
I see a rather mild asymmetry which may be the result of some asymmetrical swelling or some minor interference of lower lip muscle or nerve function. This should go back to normal within the next 4 to 6 weeks which is probably what your plastic surgeon told you. One week is very early within the recovery and shortly you should be able to enjoy the benefits of chin enhancement.
This will likely self correct after surgery and surgical trauma subside. On another note, I observed in your photo that you have gingivitis, so you may want to visit a dentist.
Smile will return to normal after chin augmentation
With this surgery, there can be a temporary problem with the sensory or the motor nerve to the lower lip. You have a motor nerve problem, but it won't be a problem for long. An injury to the nerve at the level of the chin implant will surely return to normal. It may take a little while, but it will be fine.