I'm not sure if it will be different from different countries, but I went through this process over the weekend in Hong Kong.
First, they asked me to text a certain string of characters to a designated number using a HK-based mobile number. You then receive a code that you enter while also signing into your Apple ID.
Following that, I believe they asked for my full name, pick-up time (NovemberWhiskey is correct, you choose half hour increments instead of one hour for the iPhone 6), what type of government ID you would present during pick-up and the document number.
After that you should receive an email stating that your reservation has been confirmed and asks you to be on time since if you are late, they may refuse to sell you the watch. I got there early because of that, around 50 mins early, and they actually allowed me to start my session even though my time slot was 4:30pm and it wasn't even 4pm yet.
Pretty straightforward and less stuff to go through than the iPhone 6 reservation process, since no colour, storage size or carrier/unlocked selections had to be made.