When thinking about the nervous system, most people think about the brain and spinal cord: the CNS. However, the enteric nervous system controlling gut function has another ~500 million neurons.
Among other things, these neurons coordinate the peristaltic contractions of the gut which act to move food through the digestive system.
Peristalsis is under the control of several neurotransmitters and neuromodulators. One of these is acetylcholine. One of the two major classes of acetylcholine receptors are known as the nicotinic acetylcholine receptors (nAChR). These receptors are so-named because nicotine, found in cigarettes, is a strong agonist for them (although their typical endogenous agonist is acetylcholine).
There is good evidence that nAChRs are involved in gut peristalsis. Antagonists of nAChRs reduce peristalsis Kadowaki et al. 1996 and agonists increase peristalsis Blank et al. 1999.
Therefore, you can expect that cigarettes, which contain nicotine, would increase gut motility and therefore make bowel movements more likely or more imminent. Of course, there can also be daily cycles of bowel function, so you cannot easily link smoking in the morning to a specific subsequent bowel movement.
Blank, E. L., Greenwood, B., & Dodds, W. J. (1989). Cholinergic control of smooth muscle peristalsis in the cat esophagus. American Journal of Physiology-Gastrointestinal and Liver Physiology, 257(4), G517-G523.
Kadowaki, M., Wade, P. R., & Gershon, M. D. (1996). Participation of 5-HT3, 5-HT4, and nicotinic receptors in the peristaltic reflex of guinea pig distal colon. American Journal of Physiology-Gastrointestinal and Liver Physiology, 271(5), G849-G857.