In [K Imada et al. Nucl. Fusion 59 (2019) 046016] a new 4D drift kinetic nonlinear theory, valid in the limit of a low beta, small inverse aspect ratio, circular cross section, toroidal geometry, to describe the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is derived. In [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] this theory is reduced in a low collisionality limit, which allows a dimensionality reduction to a 3D problem to efficiently resolve the collisional dissipation layer in the vicinity of the trapped-passing boundary. [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] adopts an improved model for the magnetic drift frequency, which reduces the threshold magnetic island half-width from $8.73 \rho_{b i}$, where $\rho_{b i}$ is the trapped ion banana orbit width, to $1.46 \rho_{b i}$, making it in a closer agreement with experimental observations for the large aspect ratio tokamak equilibrium. In the present paper, the theory is extended to a high beta, arbitrary tokamak geometry to capture the plasma shaping effects on the NTM threshold physics with the focus on the non-zero triangularity discharges that are known to have a strong impact on the plasma MHD stability. First, it is found that the higher triangularity plasma is more prone to NTMs which is in agreement with the $2/1$ tearing mode onset relative frequency measurements in DIII-D. Second, the NTM threshold dependence on the tokamak inverse aspect ratio obtained in [A V Dudkovskaia et al. Plasma Phys. Control. Fusion 63 (2021) 054001] is refined and extended to a finite aspect ratio limit. Third, the NTM threshold dependence on poloidal beta is obtained and benchmarked against the EAST threshold island width measurements.