Previously, the conjugate gradient method combined with fast Fourier and fast Hankel transform (CG-FFHT) was developed to solve an integral equation for borehole induction measurements in axisymmetric media. In the CG-FFHT method, the regular fast Fourier transform algorithm uses equal-spaced sample points, while the fast Hankel transform algorithm requires the sample points to distribute uniformly on a logarithmic scale. The uniform grid limits the utility of the CG-FFHT method since it is not the most efficient way to discretize a problem, especially when there are fine structures in the geometry. These limitations are removed in this work by the use of the newly developed nonuniform fast Fourier transform (NUFFT) and nonuniform fast Hankel transform (NUFHT) algorithms. The combination of the CG procedures and NUFFT and NUFHT leads to the CG-NUFFHT method. It retains the computational efficiency of the CG-FFHT method but has the flexibility of a nonuniform grid. Excellent agreement is shown between the CG-NUFFHT results and those from the numerical mode-matching method.