The Police did NOT give up. They carried out all the searches they reasonably could, using all their experience of similar cases and employing a lot of equipment, and then withdrew to await developments. Which duly developed. They knew she was in the river but sometimes they have to wait for things to take their course.
The point is that in the UK someone is reported missing every 90 seconds. Just under 1,000 people per day, most of whom have families other concerned persons who expect the police to immediately pull out all the stops to look for them. However, cases are prioritised, primarily according to the vulnerability of the missing person, and resources deployed accordingly. It therefore makes sense for the police to redirect some search and enquiry resources to other cases while awaiting developments in others.