What would be the quickest route to become an officer in the Navy and how would I follow through with it?

Your best bet as a sailor is to get to sea, complete all of your qualifications well ahead of time, get top evaluations, get at least one warfare pin quickly, and then apply. You don't have a college degree so you'd need to go to a college program, and there are several.

I applied to NROTC while I was in Nuclear Power School, back in the dark ages, and was accepted. But times are different now.

If you want to be a warrant or LDO, you need to be an enlisted sailor and go up through the ranks first.

Of course, one has to ask, if your goal was to be an officer all along, why didn't you go to college or apply to NROTC and the Academy straight away?

OPNAVINST 1420.1 SERIES is your best friend.

Get your Bachelor's degree in a specialty that is useful in the military such as buisness management or accounting with a potential diversion to NROTC. That would increase your chances considerably. As an E-3 in A school depending on your enlistment you may want to start the associate's degree part while you're out in the fleet then when your hitch is up go directly for your BA or if you work hard enough you may come out with your MA with the Navy paying your tuition for service as a Lt.