This depends on the seriousness of the lateness, and whether this has happened before.
Consider a trivial lateness: a group of friends are gathering to have coffee and snacks. Most arrive on time, and talk to each other, ordering food and drinks without waiting. One arrives late, apologizes, and is really the only one to suffer from the absence. Here you can accept the apology without a second thought. There was no harm done.
Now, a medium-severity lateness: late to work, late to a meeting at work, where others can do some work without the latecomer, but run the risk of needing to repeat things, or of delaying important things, so they are waiting or they are doing lower priority work. Late to important family occasions so you miss the first act of your sibling's play, or to some sporting events, can also fit this category. Here, assuming you're in charge (you're the person's boss, parent, or otherwise in a position to correct them) you can accept the apology but you must communicate clearly that this is unacceptable. The person may just smile "sorry I'm late!" in a way that suggest they aren't really sorry at all, or they may issue an appropriate statement of regret along with a good explanation, but nonetheless the acceptance must include a reminder that this lateness cannot happen again. Something like:
That is unfortunate. I presume you know how to ensure it won't happen again? (Wait for response.) I'll forgive this occasion.
You want to leave an impression of "I might not forgive the next one" or even "I will not forgive the next one" not "being late is ok here."
If you're not in charge, you can still try a response like this, but know that they might not pay much attention to it.
Now a more serious one, either a repetition of a previous lateness, for the same reason, or being late in a way that inconveniences customers (an instructor arriving late for class, the person who was supposed to open a store or restaurant not being there before opening time, a doctor, nurse, or firefighter arriving late to relieve the previous shift) or is dangerous, or is so socially-unacceptable that all involved assume the person must have been in an accident or something, and are distressed worrying about them. Here I would not accept the apology. It's a great use for
We'll discuss this later.
In most cases that would be the entirety of my response to their apologies for lateness. It indicates strongly that something else is more important than their lateness now, but the lateness cannot be overlooked, and it does not including accepting the apology even a tiny speck. I have literally said nothing more than those 4 words to someone who was very badly late in an important context.
Then you have the wedding or the meeting or get the store open or whatever needs to be done right then, not delaying any further, and after the crunch is past, your top priority is meeting the person and letting them know just how unacceptable (up to and including "you're fired", or perhaps "next time, you're fired") this is. Again if you're not actually in charge of the person, you can still use this "discuss this later" line, and have a heart-to-heart later about how awful it was that they were late to the funeral/wedding/awards ceremony, and the impact this has on their family and/or friends.