You don’t need to be a student to get a youth mobility (aka working holiday) visa, though you would need to wait for your country to be added to the list. But supposing you had a WHV, your status as a student would (apparently) just be an extra detail employers could use to avoid minimum wage etc. (I’m not convinced that this practice is legal.)
More info about WHV’s here:
There is also such a thing as an internship visa, as explained in this thread:
Also, some kinds of volunteer work are now permitted, but apparently not for tourists: