The heat transfer simulations of turbine blades with internal cooling are faced with so many uncertainties, of which some originate from the secondary air system, including the inlet hot gas temperature and pressure and the cooling side boundary conditions, and the blade material. The main objective of this work is to carry out a suitable sensitivity analysis on a specific novel turbine vane to improve the thermal performance of its internal cooling system and to quantify how the uncertainties on the designed/calculated values can desirably/undesirably affect the maximum blade surface temperature, which can consequently affect the gas turbine engine efficiency. Furthermore, the sensitivity analysis is carried out to find the effects of uncertainties of a number of key parameters on the resulting blade temperature distribution. To arrive at trustworthy conclusions, the conjugate heat transfer (CHT) method is used to analyze the heat and fluid flow behavior. This work suitably develops a CHT-based method/solver to perform the proposed study. This method/solver uses a segregated iterative procedure, in which the outer hot gas region is simulated using the computational fluid dynamics (CFD) solver, the flow passing through the connected internal cooling passages is calculated using a 1D correlated-based solver, and the vane conduction is predicted using a 3D finite-element solver, which are fully coupled. The results show that the cooling channel wall temperature has a direct impact on the convective coefficient magnitude; especially in lower temperature regions. As a novel contribution, this work takes into account the cooling wall temperature influence on the 1D code calculations. To implement this, an artificial neural network is suitably trained to predict better convective coefficient. The results of this developed CFD-CHT code are validated against experimental data available for a benchmark vane. Eventually, the sensitivity analysis is carried on the present specific novel turbine vane.