This was a project for our two and a half year old nephew. We wanted to buy one but was shocked by the price for a decent wooden model. The key was to find a set of smooth rolling wheels and then design the dimensions around the diameter and axles. The seat is adjustable both up and down as well as front to back.

Based roughly on BMX geometry

Step 1: Materials

A lot of this project can be made with leftover bits and pieces

1/2" Baltic Birch Plywood- main frame
3/4" plywood- front fork supports and seat post
Large Carriage Bolt- steering column
Small bolts- attaching sides of frame
3/4"x14" dowel (old broom handle)- handlebar
wood screws- seat attachment/adjustment
Polyurethane
12" wheels (from $7 thrift store bike )

Step 2: Cutting Out the Parts

Picture shows main pieces on a square inch grid for reference. On the finished product I modified the seat a bit and shortened the front fork pieces.

Sketch out one side of the frame and fork on 1/2" plywood
Cut out with a jigsaw (I used a large drill bit to make smooth corners for the inner cut outs)
Trace the cut out pieces and repeat
Round all edges with router and 1/4" bit
Sand

Step 3: Front Fork and Steering

The two sides of the frame form a wedge. With both halves touching in the front, determine how wide the back will be based on your rear axle. The tires I found required an 11 degree taper.

Cut a wedge piece to fit in the inner gap of the two halves. (This will help keep the steering column from sliding around)
Glue and bolt together.
Drill a hole for the steering column

Cut two rectangles out of 3/4" plywood the width of the fork pieces and the length of the inner axle
Drill holes through both rectangles to fit carriage bolt for steering
Sandwich body of bike between rectangles (with washers) and bolt.
Cut off excess bolt with hacksaw

Attach front wheel to both fork pieces with slots/holes to fit axle
Determine how high up the front fork the body will rest without rubbing on the tire
Attach with wood glue and/or screws

Cut dowel to 14"
Drill appropriate wholes through and glue in handlebar

Step 4: Seat and Post

For the seat:

Use 3/4" plywood to build an appropriate seat post. I chose an angle that allowed the seat to slightly tilt back when installed
Use 1/2" plywood to build a seat (I used the same 11 degree taper as the frame) A small mud flap could also be designed at this point
Round the edges with the router
Screw seat to post (don't glue, it can be adjusted to fit the child)

Attaching the seat:

Build two spacers out of 2x4 scrap. For my bike the 11 degree overall wedge meant both blocks needed 5.5 degree angles cut on the mitre saw. Adjust the width of these pieces by dryfitting with seat post in between. Once snug, glue both spacers to the inner walls of the frame. Do not glue the seat post, use screws so it can easily be raised or lowered as needed.