First find a piece of straitish knot freeish wood about 3 and a half inches wide (90 mm) or more and half an inch thick or more and about 14 inches long or more. I got mine from a section of a plank taken from a shipping palette. A longer plank will give a longer ukulele. I have made a 2 foot long one (which is a Tenor Ukulele) using a virtually identical method and a standard soprano as well.
You can see how they sound here
First is the sopranino tahitian style tuning (apologies for the fat fingers)
Second is a tenor Uke tahitian style tuning.
Third is a regular tuned soprano uke all made using roughly the same method.
Step 1: Mark out your Uke
The scale length here was going to be 20 cm but finished up being 22 cm,or around 8 and a half inches.
Step 2: Excavate out the back
Now turn the Uke over and mark out the area that needs to be hollow. Then have at it with your preferred method of excavation. For this small one I decided to go manual and used a chisel. Remember a plane is just a chisel held at a particular angle with a guide to stop it from taking too much off at once, working patiently from either end with the chisel and occasionally using a sharp knife to clear the ends and work on inside of the sides.
It is much easier to do this with a router.
Even I was not perfect and had to glue a crack back together.
If you don't feel up to the hollowing task you can just cut a hole and use something else as a resonating membrane. Heat shrunk PET bottles work magnificently.