The caps and smart uniforms at urban zoos like Bristol & London were mainly for 'best' wear. They had overalls for all the daily cage cleaning/food prep routines etc. Back in the day, after lunch, one or more staff from each animal house or section would change into their uniform to go 'on duty' -standing in a visible position near the main exhibits during the busy afternoon period to prevent public feeding of animals, answer any questions and generally 'police' their area. Junior keepers usually didn't get the uniform though, if they needed to stand-in 'on duty' occassionally they would often just wear the cap as their mark of authority! Of course you always saw keepers in full smart uniform for things like royal or other important visitors and other press events. Those old Pathe news clips are typical of that era.
Head keepers and overseers who didn't do the hands-on work tended to wear them fulltime though.
The first place I ever saw keepers in less formal wear was Chester, where even back in the late 60's they wore a more informal-looking lightweight coat/smock type thing with Chester Zoo emblazoned on the lapels. A lot has changed since those days- some of the bigger zoos have been through several different dress codes since then.