Just in case some insomniac British OAPs are reading these boards now… 
I was watching some old 1950s/early 1960s Alec Guiness and Peter Sellers movies I recorded on TCM.
In a number of the movies, but most notably in ‘The Lavendar Hill Mob’ (where they hijack gold bullion and smuggle it out of the country as replica Eiffel Tower souvenirs), there are several police chase scenes where the Police cars, instead of using sirens or the earlier 2-tone horns, sound bells (sounding like a cross between a bicycle bell and a old phone bell).
So, did the police really use such bells instead of horns/sirens in that era? I always thought they used the 2-tone horn - ‘blues and twos’ - but maybe Ealing or British Lion Studios couldn’t afford that sound effect back then.
Time to take my one bump on this, since the UKians should be up and about by now… 
Anyway, I tried to find some clips of this on YouTube, but the only clips I could find from the ‘Lavendar Hill Mob’ is of Audrey Hepburns cameo in the film, which is nice but not germane to this discussion of 1950s Police cars and bells (as depicted in some films of the era)
I’m not quite old enough to provide direct evidence, but police cars with bells certainly seems to have been the standard in the 50s and earlier, judging by many films other than The Lavender Hill Mob. Here’s The Blue Lamp, from 1950.
[edit] OK you mentioned other films, but I very much doubt that the bells were a money-saving effect. I don’t see how it would have saved them any significant money.
Yes, police cars of the period (and, indeed, ambulances and fire engines) did use bells, known as “Winkworth Gongs” from the name of the company that manufactured them.
I remember seeing film of the Blitz with fire engines that had bells. Didn’t know they lasted later…
Thanks, giving the actual name of that bell allowed me to google around and find more information, including that those bells were used by some UK forces into the 1990s - and as may be guessed, were rather ineffectual by then due to improved (i.e. louder) automotive stereo technology of the 1980s:
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Wow, I can not even imagine US law enforcement type using that kind of bell - we’d think it was some daft bicyclist trying to be annoying. That’s why I threw out some lame possible excuse for Ealing studio using a bell sound effect instead of, say, a two-tone horn - it just kinda blew my mind…
Well, it has to be said that they were a lot louder than a bicycle bell: nearer to a fire alarm.
Ambulances used these until much later. I think the Auxiliary Fire Service engines pressed into service in the fireman’s strike still had these, initially.