Another thing which my teacher did which helped me was to get me to play only a tiny snippet - sometimes only two notes - several times over, to get a given point across, exactly as should be happening when practising at home. Difficult to do with very tiny sections without seeming patronising when early on in the learning process, but if you keep the sections smaller than makes musical sense on its own, that takes some of the pressure to produce music out of the equation...Then, if I started to appear flustered, when playing those two or few notes in a bigger context, he'd say, "well, you've demonstrated you can actually do that bit now, so I'll leave you to get on with it at home". That again takes pressure off.
I get uptight if I think I'm being tested, and that is the case even yet: at a recent masterclass-y sort of lesson with an eminent recorder player, I found that when he wanted everything to be perfect, and wanted every perfect thing to be linked up to everything else (two separate processes

), I also got nervy and began to make more and more mistakes, so your student has my sympathy. I was desperate to show that I could respond to what he was trying to get me to do, but trying to do it, instead of just letting it happen, began to get in the way. If it's any consolation to your student, six years ago I would have been almost as extremely uptight as he sounds to be if I even suspected someone might be overhearing my playing, but through that first recorder teacher, I've got to the point of being able to play, unaccompanied, to an audience, to examiners, and even to Philip Thorby.