I am pleased to inform you that you can stop looking; it is well known that such a function cannot exist. The point is that URMs are more powerful computers (in a sense that can be made mathematically precise) than the primitive recursive functions. That is, programs on URMs can simulate any primitive recursive functions; i.e. every primitive recursive function is URM computable.
More powerfully, there is a universal machine which can simulate any algorithm (according to the Church-Turing thesis).
Edit:
Since you updated your question to remove the requirement that the function be primitive recursive, I am updating my answer.
The function constructed in the theorem you cite (Theorem 2.6 of Chapter 4 of Cutland's book) is a good (and I would even say, 'concrete') example of an non-computable function.
Note that most functions from $\mathbb{N}$ to $\mathbb{N}$ are incomputable. However, (roughly speaking) most such functions that arise in mathematics are computable.