I've built 15 or 16 now, using primarily hand tools. If you're prepared to do stuff the slow way, you could get away with the following:
A tenon saw for most of your sawing, or a Japanese pull saw.
A thin-kerf saw for fret slots - you could thin down a hacksaw blade, but the Zona *thick kerf* dovetail saw (around .23 kerf) is worth the £12 or so it costs and is good for fine work like cutting the bridge blank.
A card scraper and burnisher - you can thickness with this, though it is slow work. Learning to sharpen it takes time.
A chisel - probably half inch.
A drill for the tuner holes
Quite a few clamps (say 6 x 3 inch G clamps), and the spring clamps sold on UK markets at £1 a dozen are useful as well.
Sandpaper in various grades and scrap wood (B&Q pine would do) to make sanding blocks. A bit of broom handle warpped in sandpaper is good for curves.
Sharpening stuff - diamond stones work, or wet and dry paper attached to a flat surface such as a piece of MDF.
MDF to make moulds and jigs.
Screws and screwdriver (or nuts, bolts and spanner) to hold moulds and jigs together.
A half-round file - this does frets, and can be used for neck shaping.
At some point you will want a knife, though you could get by without one.
With this lot, I built my first couple of ukes.
Then I bought planes, more chisels, more clamps, more saws, and ... you get the idea.