I'm assuming that you have an existing ("main") wordlist and the downloaded ("NYT") wordlist and that you have ranked the former to some degree -- and the latter perhaps not at all. I'm also assuming that you have Crossword Compiler which contains Word List Manager... under the Words tab. (Most of my instructions below will be run from that Word List Manager).
Here's what I would do. First, make backup copies of both wordlists -- just in case you make unchangeable adjustments that you don't like. Within the Word List Manager menu, select the desired wordlist, then click on the Word List tab. From there you can select Duplicate/Filter list... The menu that pops up will allow you to do many wordlist manipulations. First make the backup copies by simply entering something like MainBACKUP and NYTBACKUP in the field which says Name for new word list. (Next, you can return to that menu for a wordlist that you want to change and play around with it to see what capabilities are possible).
The main issue with merging wordlists is that one list may be suitably ranked (e.g., your main list) and the other might have no rank distinctions (e.g., a freshly downloaded version of the NYT list). So, if the NYT list comes with all words ranked at 50, then that's where those words will reside relative to other words in your main list when merged. If you changed that number to, say, 90, then the NYT words would swamp the top end of your fill menus (once merged). If you changed it to, say, 10, the NYT words wouldn't show up in the fill menu unless the threshhold fill rank were lower than that number. So, you have to select a nominal middle rank to add those words so that you can have access to them when merged into the main list. Perhaps a rank of 60 would suffice? (The other aspect is that you'll want to rerank these words later to make the fill process menus more functional).
Much of this you have to figure out by playing around with wordlists, but an important function within the Word List Manager... menu (under the Word List tab) is the Add other lists... menu. This is all very complicated to explain -- with all its nuances -- so you'll have to do the best that you can with these instructions.
Perhaps the best option is to get a mentor who will supply a partially ranked wordlist which already has the NYT words in the first place. The alternative is that you'll have to rerank many many words that someone else has already put the effort into ranking in the first place. Something to consider.