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Does the weight of ammunition add to the weight of a wielded weapon? I would expect it to, but rules are weird sometimes. It might be a case of "This weapon weighs X lbs, no matter what you load it with."
Example: Longbow is 3 lbs. Arrow weighs 0.15 lbs. When you're shooting, is that bow now 3.15 lbs, or still 3 lbs? Say that the arrow was in a haversack or something so it had no weight until you drew it.
Since encumbrance is "up to x lbs", you might be at light encumbrance until you load your bow, be at medium to fire, then back to light once you shoot.
I can't think of any other cases where the weight of the weapon would matter, but there might be some.
There are no specific rules on it. If you want to get into that kind of nitty gritty detail for every single item feel free - but it seems excessive overkill to suddenly change someone's encumbrance level because they added 0.15 lbs of weight to their carried load temporarily.
I'm not sure why you'd need it added as part of the weapon in any case. It really it just the two separate weights.
The weapons weight, and the arrows weight. The arrow being nocked doesn't change anything for either weight.