BBM - I'm not sure what they could have done differently? They ascertained the most likely thing was a (likely accidental) fall into the water. They checked witnesses and CCTV and car cam footage etc for possible signs of NB leaving the area any other way than the river or signs of an abduction. They've talked with family and friends about health and state of mind, and tried to find out if anything else was happening in her life.
It didn't make any major difference to calls to locals who might have been in the vicinity, or people driving through the area, on that morning to know anything about her other than general description and to report anything 'odd' or unexpected that was observed that morning.
What else could police have done to prevent the hysteria? Found her body sooner? That would have helped, but I don't think the time delay was for want of trying or for want of keeping an open mind as to what may have happened.
You can't fully control the public opinions and sometimes the more you try, the more it backfires.
I think the 'hysteria' aspect is more about the social context of our times with social media and hyper-awareness of crimes against women, and opportunities for huge numbers of people to get together online to discuss things that have a 'mysterious' aspect to them. I also don't think there's anything new in the huamn desire for police to produce results 'pronto' and a fear that if there aren't immediate results that that could indicate failings or shoddy work.