After the structural changes, I decided to texture the box to give it a more concrete-like appearance (and a more interesting surface) and also weather it to make it look old. The finish before was smooth and semi-gloss, which was rather featureless and boring to look at.
The next stage was to prepare it for painting, by removing anything that was not to be painted. I also sanded down the whole box with an orbital sander to give the texture some better purchase on the semi-gloss paint. I also took the opportunity to reposition the hole for the lock onto the middle crossbeam. I filled the old hole with some dowel and wood filler. I covered the window frames, because they're not supposed to look like concrete...
I sprayed all the visible surfaces of the box with spray texture - a product normally used for texturing wall-board on the interior walls of houses (at least in the US). After that, I painted it with primer to give the texture some strength.
After that I dismantled it to paint it - to ensure that all the cracks have blue paint rather than white. I then reassembled it to apply the weathering, as I wanted to avoid strange discontinuities in the degree of weathering applied.
You also need to get enough paint onto the cloth that it wipes on rather than wiping off. You also need to keep the cloth wet enough, as the moisture does end up on the box. A tube of artists acrylic is just about enough for one police box. The picture to the right has a good closeup of the dirtying if you click on it.
And here it is with all the upgrades and weathering applied, back in its usual location.
And a couple of closeups:
Here are a couple of nice shots that high-light the windows: