Overhead high-voltage transmission lines (HVTLs) extend over diverse geographic regions under uneven weather conditions, where sag and wind-induced conductor motion poses serious issues for uninterrupted power distribution by HVTLs. Sag reduces the ground clearance of the line, whereas conductor motion may result in phase to phase short circuit, and damage to support structure. A real-time estimate of these factors is deemed important to develop requisite prevention measures. In this paper, we present a novel method to cater for sag and motion estimation in real-world scenarios. Our method can detect a change in sag and conductor motion, whether it is symmetrical or unsymmetrical to one or more conductors. We require uniaxial magnetic-field (MF) sensors mounted at support tower, equal to conductors, and synchronized with current-transformer (CT) data at substation. The sensors provide noncontact MF measurements, which are time stamped and transmitted to the substation. The data are then simultaneously processed along with electric current readings from the CT. For processing, we developed a unified algorithm that first distinguishes variation in sag from motion in conductor. Then, it estimates the corresponding sag or motion in any of the conductors. The method is verified first by numerical simulations, and then with a scaled setup in laboratory. Different test cases were identified by the algorithm. Error remains <= 1% for unsymmetrical sag estimation. In addition, trajectory of a conductor in motion is retrieved. Where the root mean square error between actual and retrieved conductor amplitude was only 0.0833 for the observation window of 200 ms.