Model 1816 muskets were standard issue during the Mexican War, and were used early in the Civil War, being gradually replaced as rifled muskets became available. Most had been arsenal converted to cap, although some flint guns were still in use, particularly in the South, where most anything that would shoot was valuable. The most common load was "buck and ball", a cartridge with a ball and three buckshot.
I wouldn't be doing anything to the gun. You could get a percussion conversion hammer, but that isn't how your family used it. It is actually in pretty decent condition. The varnish hasn't harmed it; if anything it has preserved it.
I would not replace the nipple and try to shoot it. It would be easy enough to remove the remains of the broken nipple, freshen the threads and install a replacement, but there is no point in doing that. It is an old family gun, enjoy it as it is.
You should do everything you can to record the history of the musket.
I'm new to researching this, and am of course mostly guessing, but I am thinking that this gun wouldn't make it all the way south to the Mexican war, then all the way back to NB. I am open to any thoughts or corrections about that.
As to the civil war, I was just picking the odds, as it seems from what I read, most of the model 1816 guns were converted to percussion, with rifling added, if the gun passed for those changes. I guess its possible this was used as a flintlock, then later sold and ended up in the family here, where it was later converted to cap lock. However, given that he was born in 1800, came to NB from Ireland about 1820 (another interesting story), and died in 1884, it would seem that he would be getting older for purchasing this firearm at the time after the civil war. By then his family was grown and having their own kids, and the farm was divided between his 2 sons, and he was living with one of them. So it would seem that the gun would have been purchased by one of the sons by that time, yet the family history is quite clear it was his. Admittedly, its hard to reconstruct history (but fun trying).
I have no intentions of changing anything on the gun. Adding the hammer piece of course doesn't change anything. Even what we've determined to be a hammered nipple is part of its history, and I wouldn't touch it, though like you say it wouldn't be that difficult to chase new threads.
As to shooting it, I value my fingers too much. That's why I was thinking of buying the reproduction gun as I mentioned in the OP, so I could experience shooting what he shot maybe as much as 175 years ago.
I am basing the 175 years on what mooncoon said, that is percussion caps came into use about 1830. But given this one was made in 1827, I don't think he would or could have acquired it that early. I guess I really don't know when he got it. Maybe sometime between 1840 and 1870? Maybe he used it as a flintlock for a while??