We present a first-principles computational scheme for investigating the ballistic transport properties of onedimensional nanostructures with noncollinear magnetic order. The electronic structure is obtained within density functional theory as implemented in the full-potential linearized augmented plane-wave method and mapped to a tight-binding-like transport Hamiltonian via noncollinear Wannier functions. The conductance is then computed based on the Landauer formula using the Green's function method. As a first application, we study the conductance between two ferromagnetic Co monowires terminated by single Mn apex atoms as a function of Mn-Mn separation. We vary the Mn-Mn separation from the contact (about 2.5 to 5Å) to the far tunneling regime (5 to 10Å). The magnetization direction of the Co electrodes is chosen either in parallel or antiparallel alignment and we allow for different spin configurations of the two Mn spins. In the tunneling and into the contact regime, the conductance is dominated by s-d z 2 states. In the close contact regime (below 3.5Å), there is an additional contribution for a parallel magnetization alignment from the d xz and d yz states which give rise to an increase of the magnetoresistance as it is absent for antiparallel magnetization. If we allow the Mn spins to relax, a noncollinear spin state is formed close to contact due to the competition of ferromagnetic coupling between Mn and Co and antiferromagnetic coupling between the Mn spins. We demonstrate that the transition from a collinear to such a noncollinear spin structure as the two Mn atoms approach leaves a characteristic dip in the distance-dependent conductance and magnetoresistance of the junction. We explain this modification of the spin-valve effect due to the noncollinear spin state based on the spin-dependent hybridization between the d xz,yz states of the Mn spins and their coupling to the Co electrodes.