I can't speak as to what to expect, but what I'd tell you is to try to detach from your nervousness. He, like any other player, is just a person. He plays ukulele well, and has a lot of experience teaching. But beyond that, he's just a guy. He seems to be a nice guy and a kind person...so that's an added benefit.
I'd have something ready to play that shows what you can do, or what you need help with, and also have the ability to share that with him (if it's a printed resource). Unless you're starting weekly lessons, and then the time might be more organized in terms of how the lessons are going to progress.
A 15 minute lesson does seem pretty short, however...I would think 30 minutes would be a standard length for a lesson. My adult voice lessons (post doctorate) used to be an hour.