Step 1: Conduction speed rises with fiber thickness and with myelin, so the slowest impulses travel along the thinnest unmyelinated fibers.
Step 2: Those thinnest fibers are the C fibers, measuring under one micrometre across and lacking myelin, with velocities as low as $0.2$ to $2$ m/s.
Step 3: The autonomic fibers that belong to this slow C class are the postganglionic fibers, which is why they conduct most slowly of the options listed.
Step 4: Preganglionic autonomic fibers are myelinated type B and faster, somatic motor fibers are large fast A-alpha, and lumping all sympathetic fibers together still includes the quicker B fibers. The slowest, therefore, are postganglionic autonomic fibers.
\[\boxed{\text{Postganglionic autonomic nerve fibers}}\]