Royal Caribbean cruise?

Ok, here's how it works.  Let's assume your mom doesn't have a drink package and is paying as she goes.  She goes to the bar and orders a bucket of beers and gives you a few.  No problem.  No one is going to say a word unless you get drunk and stupid.  If you are at dinner, and your mom orders a bottle of wine, they MAY not have a problem with you having a glass, it can go either way. The key is, don't get drunk and stupid.  They COULD put you off the ship if you got caught, but it's unlikely.  It isn't like a US-based liquor control board is going to bust a cruise ship. 

Most cruise lines allow you to bring two regular bottles of wine on board at the embarkation port. You ask your cabin steward for a couple wine glasses, and you can drink it in your cabin. If you buy booze in a port, you surrender it when you get back on the ship and get it back the last night of the cruise. 

The drinking age in most ports is 18.  That said, the only places I've ever seen it enforced is in properties owned by US-based companies.  So you are legal at most ports.  I'm unsure how it works in Coca Cay.  You get drinks the same way you do on the ship, so again, if your mom orders two drinks and gives you one, no one will care. 

Ok, different scenario, let's assume your mom has a drink package.  Generally, you are limited to getting one drink at a time.  You can't even get a shot and a beer.  There are packages with a specific number of drinks per day, and some that are truly unlimited.  Cruise lines are very strict about sharing drink packages.  So your mom could go to a bar, and get herself a beer, and go to another bar and get you one. As long as she doesn't push it, again, no problem.  As you can tell, the potential to drink is impacted by having or not having a drink package.

All of this is based on my experience with my kids on Carnival, RCL, and Celebrity.  They were underaged on all three cruise lines, and I had no problem getting them the occasional beer.  Sometimes they could share in a bottle of wine at dinner, sometimes they couldn't.  When my kids were 19 and 21, the 19 year old used the 21 year old's sea pass card and ordered for himself (they look alike). When they were 15 and 17, my 15 year old was ordering shots of Jaeger in a bar (I wasn't around). Your money is your ID. 

You'll be able to drink in Nassau, but that's it

Double check with the cruise line's website, but more than likely the drinking age onboard and on the private island is 21.

Not on the ship.  Only in those ports where it is legal for your age

You can't drink on the ship or any property owned by the cruise line.

If the ship's home point of origination is in the USA then the onboard drinking age will be 21, if the ship leaves from another country the onboard drinking age will likely be 18.  

What you do on shore is none of the cruise line's business, they have no say on what you consume and where you go once you have left the ship.  If you are over the legal drinking age for that country then you are legally able to drink alcohol (there are businesses in such places that specifically cater to American tourists between 18 and 20 coming off of cruise ships).  Your only restriction is that you may not return to the ship with alcohol, it will likely be confiscated.

I'd be willing to be cash money that you can drink according to local laws applicable in the waters in which the ship sails. Even if not, I'm also willing to bet more cash money that if your parents acquire a drink for you to celebrate with, no one will care or say anything.  Enjoy.