If I know beforehand, I would say "I am so sorry. I shall not arrive until [time]. Something has come up that I must deal with."
If I just arrive late, it might be " I am sorry. I'm late. Traffic was appalling and I had a problem finding somewhere to park/The buses/trains were running late and I couldn't get here earlier".
If I have to cancel, I might say, "I am sorry for the late notice, but I am not going to be able to come tonight/today." A reasonably good excuse should be given [I am ill; The boiler has blown up and I am waiting for the repair man].
As to work, things like medical appointments should have been notified previously, so valid excuses for not going to work would be illness, sudden death [of a close relative], an accident, an Act of God [Flood/Fire/deep snow]. And I would say "I am not going to make it to work today because....".
Student or Learner