This paper documents a thermo-fluid-dynamic mean-line model for the preliminary design of multistage axial turbines with blade cooling applicable to supercritical CO₂ turbines. Given the working fluid and coolant inlet thermodynamic conditions, blade geometry, number of stages and load criterion, the model computes the stage-by-stage design along with the cooling requirement and ultimately provides an estimate of turbine efficiency via a semi-empirical loss model. Different cooling modes are available and can be selected by the user (stand-alone or combination): convective cooling, film cooling, and thermal barrier coating. The model is applied to attain the preliminary aero-thermal design of the 600 MW cooled axial supercritical CO₂ turbine of the Allam cycle. Results show that a load coefficient varying from 3 to 1 throughout the machine, and a reaction degree ranging from 0.1 to 0.5 lead to the maximum total-to-static turbine efficiency of about 85%. Consequently, as opposed to uncooled CO₂ turbines, a repeated stage configuration is an unsuited design choice for cooled sCO₂ machines. Moreover, the study highlights that film cooling is considerably less effective compared to conventional gas turbines, while increasing the number of stages from 5 to 6 and adopting higher rotational speeds leads to an increased efficiency.