The "British" list is a bit thin. We have creeks as well (tributaries to a tidal river that are tidal themselves) and sloughs, washes, tarns, becks, flashes and brooks, and I'm sure many others. Canals as Bettynh says are man-made waterways,(or man-modified rivers, often called 'navigations') and branches branch off canals. 'Waters' in the more southern parts tend to be lakes rather than rivers, and 'afon' exists almost entirely in the old gaelic speaking regions where it is also spelled something like 'abhoin'.
You have to remember that Britain (although over a longer timespan) has been home to people of many origins just like the US. Even the bit called England has very distinct regions. Listen to the traditional music. In the North, 3/2 Hornpipes, in the South, unless you go back pre 1600, nothing but Euro-trash (to save arguments I admit to exaggerating slightly)!