When I was researching the Mutiny I could find no definitive answer. I have since tried a few web searches without much success.
My notes (Bengal Presidency) indicate a military style uniform with police badges, summer uniform of a yellowish khaki, officers armed with revolvers and swords and constables armed with rifles, bayonets and lathi. The turban may well have reflected the religion/area customs in the same way that the army units did. I have no references to winter uniforms.
Also one photo of the Sindh Police 1857

And one etching of the Sholapore Police
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Rattrays Sikhs were raised in 1855 as an Indian Corps of Military Police.