The concrete and wood Police boxes that have been preserved at various locations throughout the City were installed between 1932 and 1938, to extend the old system. Previously, the first Police boxes in Glasgow (and reputedly in the UK) were designed in 1891 by Fire Engineer Eggar of Glasgow Fire Brigade. They were a hexagonal cast-iron box with a large lamp on top. When the phoe rank the gas light flared up and a red filter moved over the light to give a visual indication that the phone was ringing. A great bit of engineering........but.........the cast iron box they used was an export model of street toilet for Paris Oooops. We have copies of the original patent drawings for the 1891 Police box at the Glasgow Police Museum, 68 St. Andrew's Street.
The 1935 style Glasgow Police Signal Boxes (correct title) were originally red as the GPO maintained them for the police and used the same paint that they used for ordinary phone boxes and post boxes. They were painted blue around 1973, when the GPO ceased to maintain them.
There were also Police Pillars for street corners where a signal box would not fit. These pillars were a cast iron pillar with a cube (containing the phone) and a light on top. I remember laughing at the one on Wellington Street as someone had written "For thin polis only" on the side
While attached to Maryhill Police Office in the mid 1990's, I was shown a fairly hefty electricity bill for the Police box at the Botanic Gardens. As the box had been lying unused for several years, this caused a bit of a problem. I went down to the box and found that a dosser had been sleeping in it with the heater switched on!
The inside of these boxes was, as you say, very small but they were very handy. Inside they had, obviously, a telephone, desk, stool, coat hooks, heater, drawer with various official forms, books outlining the beat boundaries, photo books of local criminals and latterly a small safe containing spare police radios and batteries. They were useful for hiding out the rain/snow etc, holding prisoners while waiting for the van to collect them, interviewing witnesses, parking a police bike (vertically).......the list is only limited by your imagination.
Hope this helps