Enumerating the elements of an integer lattice $\mathbb{Z}^n$
Your additional condition requires the tuples to be listed in ascending order of the sum of the absolute value of their components. We are free to choose the order of tuples with the same sum, so I propose the following procedure:

list(k,n) lists the $k$-tuples, where the absolute values sum to $n$:

For i=n downto -n:
  Output all outputs of list(k-1,n-|i|), each prefixed by i.


list(k) lists all $k$-tuples:

n=0;
While true:
  Output all outputs of list(k,n).
  n++.