Does the sternum fuse together after open heart surgery and how long does it take ?

Yes, like bone fract: The sternum is a bone & heals like any other bone. It is "fractured" (cut) at the start of open heart surgery. At the end of surgery, the sternum is usually "sewn up" using metal wires to hold the 2 parts together. Further bone healing (fusion) occurs over the course of 3-6 months (eg, as with a broken arm). Wires hold the sternum together in that time & fuse with bone as it heals. Ask surgeon.