We apply the coupled cluster method (CCM) in order to study the ground-state properties of the (unfrustrated) square-lattice and (frustrated) triangular-lattice spin-half Heisenberg antiferromagnets in the presence of external magnetic fields. Approximate methods are difficult to apply to the triangular-lattice antiferromagnet because of frustration, and so, for example, the quantum Monte Carlo (QMC) method suffers from the 'sign problem'. Results for this model in the presence of magnetic field are rarer than those for the square-lattice system. Here we determine and solve the basic CCM equations by using the localized approximation scheme commonly referred to as the 'LSUBm' approximation scheme and we carry out high-order calculations by using intensive computational methods. We calculate the ground-state energy, the uniform susceptibility, the total (lattice) magnetization and the local (sublattice) magnetizations as a function of the magnetic field strength. Our results for the lattice magnetization of the square-lattice case compare well to the results from QMC approaches for all values of the applied external magnetic field. We find a value for the magnetic susceptibility of χ = 0.070 for the square-lattice antiferromagnet, which is also in agreement with the results from other approximate methods (e.g., χ = 0.0669 obtained via the QMC approach). Our estimate for the range of the extent of the (M/Ms =) \frac 13 magnetization plateau for the triangular-lattice antiferromagnet is 1.37<λ<2.15, which is in good agreement with results from spin-wave theory (1.248<λ<2.145) and exact diagonalizations (1.38<λ<2.16). Our results therefore support those from exact diagonalizations that indicate that the plateau begins at a higher value of λ than that suggested by spin-wave theory (SWT). The CCM value for the in-plane magnetic susceptibility per site is χ = 0.065, which is below the result of SWT (evaluated to order 1/S) of χSWT = 0.0794. Higher-order calculations are thus suggested for both SWT and CCM LSUBm calculations in order to determine the value of χ for the triangular lattice conclusively.