Step 2: Making the neck longer
THE SOLUTION : making the neck longer.
I used a block of hard wood and shaped in to fit the neck heel one side and the tambourine the other side. I used a dremel, various fillers ans sandpaper.
When doing this, you should have in mind that the neck is slightly tilted back on a ukulele, this helps getting appropriate action and intonation. Measure this on the kit you use and try to shape the wood block in consequence.
This piece of wood makes the neck 1" longer. It could be more, since the more the bridge is centered, the best the ukulele sounds.
Advice : you must take care of the wood grain. It's always better to have the grain of the block perpendicular to the tambourine. the piece of sapelli I found was too short to do this.
Since the neck is attached with a nut, the problem is cosmetic, but if you can get a piece of wood allowing this, it will look better and will be more durable.