This problem was studied by J.E. Mazo and A.M. Odlyzko in Lattice Points in High-Dimensional Spheres. If $n$ grows with $k$ faster than linearly then the volume asymptotic still applies. If it increases slower than linearly then for most locations of the center of the circle there will be no lattice points inside it. The linear increase mentioned in the OP is a borderline case.

This problem was studied by J.E. Mazo and A.M. Odlyzko in Lattice Points in High-Dimensional Spheres. If $n$ grows with $k$ faster than linearly then the volume asymptotic still applies. If it increases slower than linearly then for most locations of the center of the circle there will be no lattice points inside it. The linear increase mentioned in the OP is a borderline case. For $n=\alpha k$ the asymptotics of the number of points $N$ inside the circle is $$\lim_{k\rightarrow\infty}N=e^{ck},$$ with $c$ a coefficient of order unity that depends on $\alpha$ and on the location of the center of the circle. (For a circle centered at the origin and $\alpha=1$ the value is $c=1.418938538$.)

This problem was studied by J.E. Mazo and A.M. Odlyzko in Lattice Points in High-Dimensional Spheres. If $n$ grows with $k$ faster than $\sqrt k$ then the volume asymptotic still applies.

This problem was studied by J.E. Mazo and A.M. Odlyzko in Lattice Points in High-Dimensional Spheres. If $n$ grows with $k$ faster than linearly then the volume asymptotic still applies. If it increases slower than linearly then for most locations of the center of the circle there will be no lattice points inside it. The linear increase mentioned in the OP is a borderline case.