Welcome to the forum, On the balance of probabilities what you suggest will be perfectly feasible. However we do not know the age or the condition of the existing cable. With it previously being used for an immersion heater, which can be a significant load running regularly, it could have caused some natural deterioration of the cable insulation. I would want to double check the continuity of all conductors and the insulation resistance of the cable and the fuse box and earthing. I would think that a competent electrician could verify all of this in less than half and hour. Also please note that in most cases, to comply with current wiring regulations, the new socket should have RCD shock protection as well as overload protection.
Doc H.