I am guessing not many people have done this yet, so I thought I'd make a thread to let you know how it went.
Before I sold my old Switch, I remembered to unlink my Nintendo account. This is done from the settings menu, and is really easy. It is very important to do this though, otherwise you will need to contact Nintendo to transfer your account.
When I started setting up the new Switch, it gave me the option to link my account to the new console... However when I tried to do it, it was not possible until I had updated to the latest firmware. Once this was done all I needed to do is open the eshop and the option was there again. All I had to do was enter my email address and password and the account was linked in seconds, my friends list and activity appeared. Re-downloading my games also seems easy.
Overall it has been a simple process, I was worried it was going to be far more complicated, because Nintendo...
The only downside is that I lost all my saves. Once the paid online is up and running with cloud saves it won't be an issue, but I was out of luck. My 3 saves on Stardew Valley, with 350 hours of game play put into them, are gone... But it gives me a reason to start again!