Time of year makes an enormous difference: water volume during the height of the monsoon (July to mid-Sept) is ten or more times greater than in February and March, making the major rivers off-limits to all but experts. The water is more manageably exciting in mid- or late October to November, which is the peak rafting season, and becomes mellower (but colder) from December through to February. March through to May has rising water levels, when snowmelt and pre-monsoon storms begin to add to flows again.
Winter isn’t as chilly as you might think, since most raftable river sections are below 500m elevation, but it’s a slow time for tourism generally in Nepal, so many river operators don’t run trips then. March and April are the best months for long, warm days and excellent birdwatching. However, different rivers are at their best at different times of the year – for example, the Sun Koshi is actually quite good starting in late September – so which river you go on will depend to a large extent on when you’re in Nepal. Note that a given trip will take less time in high water than when the water is running more slowly.