As pointed out, this question really depends on how many ukes you plan to build. If you want to go into the business of making and selling ukuleles you might want one of these:
Yeah, a pretty big investment... You can use the Wagner Saf-T-Planner too. An intriguing alternative, but I've never used one. Hopefully Pete is gonna go into this more.
I have used the plane and orbital sander method which I will describe in
excruciating detail below. It is neither the ideal nor the most efficient method, but it does work and it requires only simple tools and some patience.
After you have glued your plates together so that there is very little or no visible seam, decide which side is going to be the "show" side and which is the down side. Mark the edges of the work with a "v" to show which side is which.
Now comes the most critical part of the process: Your plane blade (iron) needs to be as sharp as humanly possible (scary sharp). This takes skill and practice to achieve, but there is no way around it.
Now you will have to somehow keep your work from moving as you plane/sand it. A challenge that has a number of clamping solutions. The work needs to be on something perfectly flat. I use a piece of MDF underneath. If you don't work off a perfectly flat surface, your top (or back) will have inconsistencies of thickness (not good).
Now plane and smooth with an orbital sander (to 120) the show side of the work so that it is flat and smooth. This doesn't take long. You won't touch this side again until later. The majority of the wood will be removed from the non-show side.
Flip the piece over and go at the wood with your plane occasionally smoothing things out with the sander (80 grit or coarser). Caution: make sure sandpaper grit/pieces of wood don't get under your piece because severe damage to the show side can occur. This becomes very important as you approach your rough target thickness. Clean under your piece obsessively. Try to have a set number of strokes with the plane and a pattern that you repeat so that the wood removal is consistent over the entire piece. In other words, don't go at it willy-nilly. The downside of this method is that it is hard to get perfectly consistent thickness. But you can get pretty darn close with practice.
In a surprisingly short but boring period of time you will reach your first rough target thickness. At around .110 or .120 it is time to cut in your rosette. Do not cut the soundhole yet. If there is to be no rosette, just keep on truckin.
Continue thinning on the non-show side until you get near the thickness you want. There is no exact measurement here other than probably thinner than you think. Lots of discussion on this I won't go into. Now flip over and sand out the show side smooth to about 180. Cut your sound hole and rough in the piece about a 1/4 inch bigger than your sides.
I think that is it. That is how I do it. Hope I didn't forget anything important. Pretty easy and quick (with a sharp plane) and can even be enjoyable if you like getting up close and personal with your wood.