We consider the task of conducting underwater surveys with a sonar-equipped autonomous underwater vehicle (AUV) in environments with strong currents. More specifically, this topic is addressed in the context of mine countermeasure operations employing synthetic aperture sonar (SAS) sensors. Two complementary algorithms that allow the AUV to autonomously adapt its survey route based on sophisticated sensor data it collects in situ, while respecting the unique constraints imposed by the problem, are proposed. The algorithms allow the AUV to (i) adapt its survey heading based on the presence of currents to ensure quality data is collected, and (ii) adapt its survey route to reinspect the most suspicious objects at additional aspects. The flexibility to immediately react in situ to the environmental and tactical conditions sensed during the mission allow the most useful data for object recognition purposes to be collected efficiently. By obviating the recovery and redeployment of the AUV, as well as laboratory-based data-processing during the interregnum, the overall mission timeline can be greatly compressed and operational costs can be reduced. Experimental results illustrating the real-time execution of the proposed algorithms on an AUV are shown for a completely autonomous mission conducted in the North Sea.