My quick thoughts on the matter are this... uniforms of the mid-19th century were evolving. The Russians were actually the first to introduce a spiked helmet (Pickelhaube), and the Prussians (more accurately than Germans) adopted a similar helmet. The Russian helmet was in fact inspired by the Tatar helmets of the 13th and 14th century. As Stuart and I have also noted the air tube helmets originated in India, and those appear to have been based on the cavalary helmets used in Europe during the Napoleonic Wars.
I like to think that the spiked helmets, which were used by dozens of nations in Europe, South America, even by the United States, were not so much everyone copying the Prussians/Germans as much as a shift away from one fashionable type of headgear, the shako, to something different. Much has been made that all these nations were so impressed by the Prussians victories against Denmark, Austria and France that nations wanted to adopt a similar look. I really don't believe it is as simple as that.
One thing I see is that by the 1860s, when the spiked helmets were being introduced, is that the last of the Napoleonic Wars officers were old men or dead. The military planners in the 1860s were either very young men or missed the last great series of wars. Thus of course they wanted to leave their own mark. Likewise, the police likely wanted to keep in line with military fashion.
The colonial pattern sun helmet was being used by the guardians of the British Empire, so why wouldn't the police at home use a similar helmet, one that matched their dark uniforms. This was industrial era Great Britain where the air was full of soot and dirt, and so a black uniform would be far more ideal than a white uniform or helmet.
That's what I have to add. But I just think too much is made that the German pickelhaube inspired every other spiked helmet.