This is a very loaded question in my opinion. The OP specifically mentioned shooting a nanny in a healthy herd which should be a pre-amble to any consideration of shooting a nanny in my opinion but more so than that, there are a lot of factors that should be taken into consideration before purposefully considering shooting a nanny.
First of all, it is important to note that mountain goat populations do not follow the same population trends of other ungulates (particularly deer and elk.) Healthy deer and elk populations can both tolerate 10-20% harvest rates in a given year and the population will continue to repopulate in that specific location. When harvest quotas were first implemented for mountain goats (in most states this was in the '60's and '70's) game managers tried to manage goat herds using the same statistics they used to manage deer/elk herds which ended up resulting in overharvest of most herds to the point that many herds are completely gone or had to be closed for a decade or more to recuperate.
As other posters have pointed out, nanny goats usually do not being reproducing until 3.5-5.5 years and are usually fertile until until ages 12-15. On the flip side, deer and elk can start reproducing sometimes as young as 1.5 (usually 2.5) years and are usually fertile until 10-12 years old. Additionally, where it is quite common for deer and elk to produce twins or even triplets, especially in herds with a good environment to support a larger herd. It is exceptionally rate for goats to have twins and almost unheard of for them to have triplets. Marco Festa-Bianchet and Steeve D. Cote are considered the foremost knowledge on mountain goat herd fecundity (reproduction) and if memory serves correctly in 17 years of monitoring a herd of 100+ goats there were only 2 or 3 instances of twins being born. Furthermore they state multiple reports on herds that were not considered healthy and the instance of twins was often even lower in those herds.) For this reason alone, shooting a goat without a nanny often results in shooting the young goats from a herd and severely impacts herd health and stability moving forward.
With all of that said, there are still some major exceptions to that rule. Festa-Bianchet/Cote's research revealed something peculiarly unique about mountain goat herds. Where an established herd has a relatively set reproductive rate that is basically set herd to herd (e.g. WA compared to BC compared to AK), introduced herds were recorded with SIGNIFICANTLY higher reproductive rates, much more similar to common deer/elk herds. Herds that had been introduced in Montana (Crazy Mountains), UT, CO, NV, and OR were all reproducing at significantly higher rates than the native herds those goats were transplanted from. Even more interesting (in my opinion) was that wildlife managers could artificially maintain that high level of reproduction by setting high harvest quotas. It is for this reason that herds in states like UT and CO have a significantly higher quota per 100 goats than states like WA, ID, and most of MT. Once the herd has been established, even if the herd is in significant decline, the reproduction rate slows down and cannot (or has not yet) be manipulated to the higher reproduction rates of other introduced herds.
That all brings us back to shooting a nanny. In general most states do allow the legal harvest of either a nanny or billy in most any unit (with a billy tag.) From what I can gather, that is primarily due to the fact that it is so (relatively) difficult to distinguish the difference between a billy and a nanny that they don't want to make it illegal to accidentally shoot a nanny. But as a rule of thumb, it should be avoided at all costs shooting a nanny out of an established and/or native herd (found in AK, BC, ID, MT, WA) unless the local game department or biologist has told you otherwise. (From my understanding, Kodiak Island is experiencing an overpopulation of goats at this time? which may be resulting in the increased quotas and season lengths on that island. That may be an exception to the rule per local biologists.) On the other hand, in some states like CO and UT (which both offer a relatively high number of nanny tags each year) nanny harvest is actually vital to herd health and continued high reproductive rates (which results in high hunter opportunity.) In those states, is is a great idea to harvest nannies under F&G guideines.
Hopefully that makes sense to everyone, feel free to post any questions which I would be happy to attempt to answer as much as my (admittedly limited) knowledge allows.
