General Information
The original police boxes were mostly made from reinforced concrete and had a single teak wood door.
The Brachacki and Newbery props from Doctor Who were made of wood, but were presumably relatively heavy for location work, so in the 1980's two fibreglass props were built to the Tom Yardley-Jones design.

The new 2005 series props were built from wood again, but fibreglass molds have since been made for later use.

What you use for a replica depends mostly on whether you ever intend it to survive any significant time outdoors.
Most builders use Pine or similar dimensional lumber, and plywood for the panel recesses.
Some people have used MDF for parts of their boxes, but MDF soaks up water like a sponge and is generally not suitable for even damp conditions.

A few builders have built fibreglass boxes, which have the advantage of being weather proof and much lighter weight.

A few boxes have been made from scrap wood from skips.