Are you genuinely concerned about the reason? If so, then 'Why are you late?' is the best way to find out. However, it is very direct and if the other person doesn't realise they are late (perhaps the meeting time wasn't clear), it could be confrontational. I would say it depends how you say it: angrily, or softly and concernedly. I often use the more indirect 'What happened?', or 'Did you forget what time we arranged to meet?', or even 'Are you ok?'.

You already know it, you've used it in your question. 'Why are you late?' is just neutral English, it is used in informal and formal situations, gets to the point and contains all the relevant information.

Don't forget in English being direct can sound rude. So just asking "Why are you late?" should only really be used when you want to show you are upset or cross that the person is late. I slightly softer way of asking is "How come you're late?", better still use this common humorous variation "What time do you call this?" using a jokey pretence of indignation. Another way might be to ask "Good to see you, I hope didn't you have any problems getting here?" Or you could offer up an explanation with which will prompt the person to confirm it or offer an alternative explanation. e.g. "Was the traffic bad getting here?". It does depend on context and your relationship with the person you are asking.

To me what you wrote is best for all situations. "delay" in America implies a slightly different meaning (that you were deliberately late" or is used in reference to planes, buses, etc. "tardy" is a formal word that means late, and is usually used at school (this student was tardy 3 times this week) and often in writing only.

What is the reason for your delay? (formal) Why are you tardy? (informal)