Usually some patients don't get pains till the date,sometimes there might be wrong dates.. pregnancy is 40 weeks,we can wait till 41 weeks if there are no comorbidities like pre eclampsia and GDM.Make sure that your wife has good baby movements.
To have normal delivery,you need to consult an obstetrician ,only after examination we can say whether she can have normal delivery or not...no doctor will give a 100% confirmation of a normal delivery.
Regarding injections and tablets,depends upon the situation,for induction of labour,there are some medications, Antibiotics,etc will be given on admission
Next Steps
As I have told you can wait till 41 weeks,but I prefer go and get admitted as early as possible.