The court has approved it, but my first wife still cries every day. Should I hurry the akad nikah?

The court's permission gives you a way forward, but it does not remove the duty to treat a wounded wife with kindness (ihsan). Rushing while she is still fragile can damage years of trust. Discuss a more suitable time with her, and listen to what worries her most. Show that her home will not be neglected through a clear rota (turn of nights), maintenance and time with the children. Offer counselling sessions, and let her choose whether to go together or alone. If she has trouble sleeping or eating for a long time, encourage her to see a doctor. Being patient for a few months is not defeat; it is a sign you are worthy of carrying this trust. See guilt and staying fair.