After three decades of accumulated experimental and numerical results, a comprehensive understanding of the spatial evolution of axisymmetric turbulent boundary layers (ATBL) along long thin cylinders still eludes both scientists and engineers. While experimentalists dealt with axial alignment complexities, computationalists lacked proper inflow boundary conditions. Herein, we correct this latter deficiency and initiate an investigation of the thin cylinder turbulence under low Reynolds numbers and high transverse curvatures (boundary layer thicknesses to radius). Using the large-eddy simulation (LES) methodology, we are particularly interested in the radial propagation of the transverse curvature on the ATBL statistics. A ten-simulation matrix was constructed to examine these effects with validation against the experimental evidence. These tests investigated the ATBL maturity up to transverse curvatures approaching 2 orders of magnitude. A recently developed turbulent inflow procedure for the thin cylinder was implemented that couples a dynamic form of Spalding’s expression for rescaling the mean streamwise velocity with recycling of all superimposed turbulent fluctuations. The technique specifically circumvents intensive computations from the cylinder leading edge, and the rescaling-recycling combination minimizes the inflow turbulent regeneration length under very high transverse curvatures. After the initial transition phase in each LES computation, the respective numerical uncertainty was quantified to ensure sufficient spatial resolution within the discretized domain for resolving the energy-bearing scales of the turbulent motion. For the present low-Re conditions, the strength of the log layer steadily diminishes under continuous rise in the transverse curvature whereas the scaled fluctuating intensities elevate (except for the dominate shear stresses) with no sign towards full maturity. Each simulation reveals a boundary layer thickness that grows downstream by a factor of 7 relative to the momentum thickness with a linear influence of the transverse curvature on the wall-shear stress coefficient.