Prior to the turn of the 1900's, European countries varied between having local or state police departments or national para-military police.
In Europe police did tend to always be armed, for a number of reasons.
One was the pistol was more a badge of office and authority. The policeman represented the king or ruler, and as such, following ancient tradition, the local symbol of the king would be armed.
For another, the policeman was the first defense against rebellion and needed to be armed to maintain control.
The pistol as a tool of actual law enforcement wasn't what it was in America.
Policemen seldom had to shoot criminals.
Up until the late 1890's most European police were armed with larger military type revolvers, often the same as the countries military were armed with.
This was in keeping with the para-military aspect of police.
Since most countries had a proprietary military revolver, that's what the police carried.
Starting in the late 1890's the automatic pistol began to become available and European police immediately began to convert to the automatic. Some like Germany and Switzerland continued to carry the same pistol as the military, in their cases, the Luger.
Many countries police broke with tradition and countries like Spain and France adopted small caliber .32 or .380 automatics, most often the Browning pistol or various copies or versions.
It was during this time of civic upheaval and anarchist and communists threats that more plain clothes detectives began to appear. Where in the past almost all police wore some sort of uniform, the plain clothes detective was better able to remain unnoticed by potential revolutionaries, and they were issued with the small .32 and .380 autos as befitting a non-uniformed officer.
In most of Europe the revolver was considered obsolete as a police weapon by the end of WWI.
At the end of WWII, the US gave the German and French police large quantities of fine S&W .38 Special revolvers, often the S&W Victory Model.
The Germans discarded the S&W's as soon as Walther was able to begin producing PP, PPK, and P-38 pistols in France in the early 1950's.
The French so disliked these S&W revolvers that they almost immediately trade them off and bought very inferior Spanish Ruby type .32 autos.
The fact that these cheap autos were none to reliable not mattering to the French cops. Better an auto then a revolver.
One of the few times the revolver made a reappearance in Europe was with some special French police who used revolvers for a short time. The first of these were built by Manurhin using frames supplied by Ruger, then revolvers entirely designed and built by Manurhin.
The German GSG-9 anti-terrorist special force used a variety of S&W revolvers for a time.
They started out with the S&W Model 60, then with some real world experience with the .38 Special and recommendations from SEAL Team six, the S&W Model 66 with a 2 1/2" barrel was used as an assault weapon for airplane cockpit take-downs.
The bottom line was that the European police bought into the automatic pistol early on, and were satisfied with calibers considered under-powered in America.
Since the police pistol was very seldom ever used they probably didn't need much more until the heavily armed terrorists of the 1960's made shoot-outs a real threat.
On the other hand, America had a long tradition of gunfighters, and the more individualist less willing to submit to authority of American's gave us a long history of police needing to have powerful, reliable pistols, and to be ready to use it fast.
Due to the early less than reliable automatics, and simple tradition, the revolver maintained it's primacy in America until the more prevalent violence of the 1960's and 70's led police to decide they were out-gunned and needed automatics.