Buying small amounts of cut wood (UK)

Do you have access to a hobby store, a large toy store, lab supply house, etc. that would have small laboratory equipment? In the US lab equipment or toy lab equipment is found in many place like mentioned.

These are also available through some online websites for store like mentioned above.

If so you should be able to buy a stirring motor that is exactly like you are trying to build. Probably less than the materials you are looking for. This is especially true if you consider the time you are spending looking for and buying the materials.

Do you not have any friends/neighbours who could do the cutting?

Like this?

Rather than trying to build a wooden base without the proper tools, why don't you look around for a good plastic box. I am sure that you could find one that would fit your need and be easily modified to suit your needs, plus it will be simple to clean up spilled liquids as they won't soak in or damage the finish.

If you build, the box, remember that you must consider the distance between the two magnets. If the distance is to great the stirrer will not work. There is a point where the magnets are close enough to affect each other but not close enough to with stand the centrifugal forces of the spin and the magnets will wobble and fly a part. (Assuming one of the magnets are free.)

The distance between the magnets is critical

Doesn't matter. Your plan won't work.

1. The fan motor is much too fast.
2. A steel pin won't spin - it will follow the magnet.

Real magnetic stirrers have a) An adjustable speed motor from 1 to about 200 rpm, and b) A rod magnet as the stirring mechanism encased in Teflon (or glass).