They did the pool and classroom work at Scuba Toys in Carrollton, TX which is a NAUI center. The instructor they had, Alan, set us up with Luis Cabanas in Cozumel who is a PADI instructor.
Cozumel - Cozumel Diving / Cozumel Dive Shops / Adventure Discovery - Luis Cabañas (Luis has it listed right on his website that he takes referrals from other agencies) They took the classroom work in February and we went to Cozumel in March. It was really for the same reasons as you. Diving here in the Dallas area in a lake in February was just too cold and gross for my wife. They did work on rescue skills with Luis. We returned after the trip to Scuba Toys and they inspected their log books that Luis signed and they had their NAUI cards shortly after. It was totally hassle free. I would imagine the place your taking the lessons in NYC has many students do the dives in another destination, for the reasons you mentioned.
Looking at the Cozumel Web Site it looks like they accept what I consider a "standard" referral...not what is described as a "Universal Referral". They say a referral "Takes 2 days and includes: Private Dive Instructor, 4 open water dives, all equipment and the PADI certification card." It looks like they follow the "Referrals from Other Organizations" requirements that are documented in the PADI Standards and follow the PADI open water training dive curriculum. They then issue a PADI Certification. This is standard stuff and should not be a problem with any PADI shop.
A "Universal Referral Program" is different, it has some specific requirements.
1. The PADI Instructor must have registered on the website prior to it going offline. This is to receive a referral number.
2. The PADI Instructor does not follow the PADI OW training curriculum, but the required skills from the Universal Referral Program Manual and Skills Sheet.
3. The PADI Instructor must issue a Universal Referral Temporary Card. How is this done if the website is no longer online? Were there paper temporary cards that were purchased in bulk?
Luis must have registered when this program was active, as there seems no way to register as an instructor since the Website has not functioned in almost 10 years.
Since there are some specific registration requirements to be an instructor in the Universal Referral Program, it is important to call to see if the PADI instructor is registered for that program. Referring a NAUI student to a PADI instructor or shop to complete their OW dives should be no issue and is documented in the PADI Standards...the Universal Referral Program with it's own set of required skills is not.
Between IDEA, NASDS, NAUI, PDIC and SSI...which agency was supposed to keep the program active? Who was maintaining the website? I ask because I think it is an interesting idea, but it no longer seems to exist in the manner the documentation I see seems to describe. Without a website, how does a referring instructor verify that the PADI instructor is registered with the program? How does it attract new instructors? Who now maintains the registered instructor database?
To the OP...you should not be concerned about starting with NAUI and completing with PADI. It happens all the time. Do not be surprised if you are presented a quick quiz or test to assess your knowledge and a confined water session to asses your in water skills prior to your open water dives. You may or may not receive a PADI Certification card upon completion depending if the instructor handles this as what I will call a "normal" inter-agency referral or if they registered for the "Universal Referral" program.