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
Rattrays Sikhs were raised in 1855 as an Indian Corps of Military Police.