As the your kid is only 3 year old fracture of bone shouldn't be too much of a concern as kids have bone growth and bone healing faster than adults. Bone healing completely should take about a month although there is risk of re fracture in the same area for another 4 months.
It is not advisable that you wait for that long to get an appointment as the bones heal fast and if there is re alignment needed, it needs to be done soon. If realignment is not done soon, either in form of splint or nails, it will lead to mal union and gross deformity. Alignment is done using a nail called K wire.
K wire is used when there is displacement of bones especially finger bones or even sometimes in fractures of long bones like femur. The procedure is done with local anesthetic itself most of times. Though if the displacement is to a great extent then they prefer for a general anesthetic. Firstly they try to realign the displaced bone in its correct position by application of traction. After realigning the bones, the doctor makes sure the finger has achieved stability in that position.