A lot of important scenes happen over or around water in Oliver Twist: the locket and ring are thrown into a river, Nancy meets Rose and Mr. Brownlow on London Bridge, and the final pursuit of Sikes is in a neighborhood surrounded by the Thames at high tide.
The water imagery is ambiguous – water should clean the neighborhood of Jacob’s Island, but it actually just carries more filth into the area. The water level is also inconstant, because the Thames is a tidal river (meaning its level goes up and down depending on the tide). Jacob’s Island isn’t surrounded by water in that final scene with Sikes’s attempted escape, but by mud, because the tide is out. And mud is kind of a sticky halfway point between water and land. It’s as though Sikes were in a sticky halfway point as well – he’s been hunted (and haunted) for so long that he feels like he’s halfway dead already.
The London Bridge could represent the same thing: it’s a halfway point for two extremes to come together. It would be hard to come up with two women more opposite than Rose and Nancy, but they’re able to meet at London Bridge, over the river Thames. Rose offers to let Nancy step over the bridge to the other side (metaphorically speaking, since they both actually live on the same side of the river), by offering her a place to stay far from her old life of crime, but Nancy refuses.
The river is also an appropriate place to bury Agnes’s ring and locket. Because she had a baby but was never actually married, she was in an odd halfway position between unmarried and married. The ring, which had her first name, but not her last, engraved on it, represents that liminality (which is a fancy word for "in-betweenness"). Since the river represents the same thing, it makes sense for Monks to drop the ring into the raging torrent.