Not a complete answer, but here's commentary from a ffdn review of Chapter 14:

Kevin S. Van Horn
7/24/10 . chapter 14
Harry is jumping to conclusions when he tells McGonagall that the Time-Turner isn't even Turing computable. Time travel simulation is simply a matter of solving fixed-point equation f(x) = x. Here x is the information sent back in time, and f is a function that maps the information received from the future to the information that gets sent back in time. If a solution exists at all, you can find it to any desired degree of accuracy by simply enumerating all possible rational values of x until you find one that satisfies the equation. And if f is known to be both continuous and have a convex compact range, then the Brouwer fixed-point theorem guarantees that there will be a solution.

So the only way I can see that simulating the Time-Turner wouldn't be Turing computable would be if the physical laws of our universe give rise to fixed-point equations that have no solutions. But the existence of the Time-Turner then proves that the conditions leading to no solution can never arise.