Since there is no region tags on the question this answer is coming from an American perspective.
Depending on the contract both you and the person who failed to relieve you could be reprimanded or fired.
There are certain jobs which have additional requirements on employee attendance. These jobs typically involve health and safety of individuals or facilities and if no one is present can result in loss of property, injury, or death. Someone from HR or management normally informs you of these requirements before you sign on the dotted line, since by signing you are now carrying legal liability if something happens and you were missing in action (MIA). These job have two common and very important rules:
- Do not show up late or miss a shift
- Do not leave if your replacement has not shown up
If you do not know if your job is one of these types you need to ask your manager. If your job is one of these, then your coworker broke the first rule and you broke the second rule. Which would put both of you in hot water and at risk of being reprimanded or fired.
How can I effectively address this situation with my employer to sort out the possible delays my coworkers may have?
The answer strongly depends on what I previously mentioned. If you agreed to show up no matter what and not leave no matter what, then you have no options to easily address this. If you cannot handle the extra burden of the possibly of having to stay for an extra shift in a pinch, then you need to find a different job that does not have this requirement. However, what if it is not?
If this situation is as dire as your manager is portraying and yet there is no legal binding on you the employee to show up or stick around, then your manager needs to make allowances and buffers to prevent these situations from arising. The easiest thing to do is have overlapping shifts. On one of my previous projects which required 24 hour monitoring, we had four eight hour shifts to cover the 24 hour period. Each shift had at least a one hour overlap with the next shift, so if someone was running late or was a no show, we had at least a full hour to confirm that they were simply running late or find someone else to cover it before the current person's shift ended.
The challenge would be to convince your manager to do this, since it will cost them more money. If you go this route you will have to be diplomatic, take ownership of the problem, and act like you deeply care about the problem and you desire to see it fixed. Even then there is no guarantee that they will do it.