A very informative thread... nice work!
A couple of things;
Britain: The colour of police uniforms was set by the Home Office and was that dark navy blue - almost black colour, as opposed to the navy blue often seen painted on miniatures. Helmet styles varied from force to force, as did the badges. 'Day' badges and buttons were silver and 'Night' badges and buttons were black.
Back then officers on night duty were permitted to carry a pistol if they felt the need. Otherwise firearms were issued to 'competent' (a very loose term) officers on the authority of a sergeant or above, at their discretion; albeit there had to be an adequate perceived need that could be explained in court.
Spain: The Guardia Civil, Carabineros and the later Guardia de Asalto, were all 'gendarmerie' (paramilitary) effectively and not so much 'police'; the first two even fell under the Ministry of Defence and all three were 'National Forces' administered from Madrid, recruited solely from ex-military personnel. They lived in 'cuartels' with their families, which were essentially fortified barracks. To those can be added the Mozos de Escuadra of Catalonia (and another force in the Basque Country whose name I forget), who stood in for the GC there effectively. These guys were always fully-armed (rifles, machine guns, armoured cars) to fight smugglers, bandits, rioters, armed militants and what-have-you; tough jobs for tough times, even by the standards of the 30s.
Local municipalities had their own 'police' forces and reserved the right to clothe and equip them as they saw fit. This led to quite a wide variety of uniforms, which varied from those that were utilitarian, to those that were perhaps more at home in 'Carnival'. These were the 'police' as we in the UK would understand them at least, recruited from, by and for, the community (although usually only answerable to the local 'big man/men' or 'caciques').
Just to add to the confusion, there were 'Summer' and 'Winter' uniforms for all that varied in detail too.