The increasingly complex oil and gas wellbore condition and the need of drilling in more challenging downhole environments motivate rising research on the drilling control system based on real-time state feedback or measurements from the bottom of the wellbore. However, due to the complex downhole condition and cost-viability, mud pulse telemetry is normally used in this industry to transmit the downhole measurements to the surface, which can cause a large data communication delay as a result of its low bandwidth and slow mud pulse transmission. Since the major drilling control is on the surface, an observer is required to estimate the real-time states of the drilling dynamics as well as the downhole condition, based on the delayed downhole measurement. In this study, we first construct a drilling system dynamics model with coupled axial and torsional dynamics. Then, with the existence of a large output measurement delay, two chain-observer design strategies are introduced for the case of slowly varying control inputs and that of fast-varying control inputs, respectively. The effectiveness of the proposed observer design methods is shown through numerical results.