Using the coupled cluster method (CCM) we study the zero-temperature phase diagram of a spin-half Heisenberg antiferromagnet (HAF), the so-called J1-J2' model, defined on an anisotropic two-dimensional lattice. With respect to an underlying square-lattice geometry the model contains antiferromagnetic (J1andgt;0) bonds between nearest neighbors and competing (J2'andgt;0) bonds between next-nearest neighbors across only one of the diagonals of each square plaquette, the same diagonal in every square. Considered on an equivalent triangular-lattice geometry the model may be regarded as having two sorts of nearest-neighbor bonds, with J2' =?J1 bonds along parallel chains and J1 bonds providing an interchain coupling. Each triangular plaquette thus contains two J1 bonds and one J2' bond. Hence, the model interpolates between a spin-half HAF on the square lattice at one extreme (?=0) and a set of decoupled spin-half chains at the other (??8), with the spin-half HAF on the triangular lattice in between at ?=1. We use a Néel state, a helical state, and a collinear stripe-ordered state as separate starting model states for the CCM calculations that we carry out to high orders of approximation (up to eighth order, n=8, in the localized subsystem set of approximations, LSUBn). The interplay between quantum fluctuations, magnetic frustration, and varying dimensionality leads to an interesting quantum phase diagram. We find strong evidence that quantum fluctuations favor a weakly first-order or possibly second-order transition from Néel order to a helical state at a first critical point at ?c1=0.80±0.01 by contrast with the corresponding second-order transition between the equivalent classical states at ?cl=0.5. We also find strong evidence for a second critical point at ?c2 =1.8±0.4 where a first-order transition occurs, this time from the helical phase to a collinear stripe-ordered phase. This latter result provides quantitative verification of a recent qualitative prediction of and Starykh and Balents (Phys. Rev. Lett. 98, 077205, 2007) based on a renormalization group analysis of the J1-J2' model that did not, however, evaluate the corresponding critical point.