Dictionary says 昼 means noon, midday, or lunch but colloquially if someone says "彼は昼過ぎに来ます" would it mean "He'll come after lunch." or "He'll come in the afternoon."?
I'd say it can be both. Since 昼 can mean (as you wrote in the OP) noon, lunch, so it depends on situation. If you are planning to have lunch with someone when you meet them, you can use "afternoon"; if you don't plan to, you can use "after lunch".
I think "afternoon" is the most commonly used, since either way he'll come to you at afternoon anyway.