We report the first-principles study of the orbital magnetism, the magnetic anisotropy energy, the ratio of the spin, and the orbital moments in nano-sized systems perturbed from their magnetic ground state. We investigate one monolayer thick films of Co, Fe, and FePt. Two types of the perturbation are studied. First, the collinear spin structure is rotated continuously between the easy and hard axes. Second, the non-collinear spin structures are considered varying in both the angles between spin moments and the direction of the net magnetization. In agreement with the experiment we obtain a variety of behaviours. We show that the magnetic anisotropy energy can both increase and decrease with increasing magnetic disorder. The type of behaviour depends on the variation of the electronic structure with increasing angles between atomic moments. We obtain the effect of band narrowing accompanying the spin disorder that correlates with the band narrowing obtained experimentally in a laser irradiated system. In agreement with this experiment we show that the ratio of the spin and orbital moments can both remain unchanged and vary strongly. We analyse the applicability of Bruno's picture, which suggests proportionality between magnetic-anisotropy energy and orbital moment anisotropy for non-collinear spin configurations. We study the non-collinearity of the atomic spin and orbital moments and demonstrate that the response of the orbital moments to the variation of the spin structure can be unexpected and spectacular.