The electronic and magnetic properties of one-dimensional (1D) 3d transition-metal nanowires are investigated in the framework of density functional theory. The relative stability of collinear and noncollinear (NC) ground-state magnetic orders in V, Mn, and Fe monoatomic chains is quantified by computing the frozen-magnon dispersion relation ∆E([(q)\vec]) as a function of the spin-density-wave vector [(q)\vec]. The dependence on the local environment of the atoms is analyzed by varying systematically the lattice parameter a of the chains. Electron correlation effects are explored by comparing local spin-density and generalized-gradient approximations to the exchange and correlation functional. Results are given for ∆E([(q)\vec]), the local magnetic moments [(μ)\vec]_i at atom i, the magnetization-vector density [(m)\vec](vecr), and the local electronic density of states ρ_iσ (ε). The frozen-magnon dispersion relations are analyzed from a local perspective. Effective exchange interactions J_ij between the local magnetic moments [(μ)\vec]_i and[(μ)\vec]_j are derived by fitting the ab initio ∆E([(q)\vec]) to a classical 1D Heisenberg model. The dominant competing interactions J_ij at the origin of the NC magnetic order are identified. The interplay between the various J_ij is revealed as a function of a in the framework of the corresponding magnetic phase diagrams.