Why do I still have mouth swelling and throbbing pain 3 weeks after implants?

The implant can be close to the nerve or accidentally came in contact with some soft tissue or sinus. Usually pain should not be there after 3 weeks. 
CBCT will detect the three dimensional position of the implant and it's proximity to any of the vital structure. Good luck

Hello
It is possible that the pain is caused by the inflammation of your gums or in its defect the nerve, nevertheless it is necessary to do radiographic studies to determine the cause
Regards.

its difficult to say anything without seeing xray. so its my advice to upload xray, so i can guide u better.