To anyone thinking of getting a new phone: make sure you back up your campsite and link it to your Nintendo Account before switching phones! It only stays linked for 7 days so make sure you have enough time to restore it. I was under the impression that your account would still be linked to your Nintendo Account but since Animal Crossing: Pocket Camp Complete came out and the game's offline now your stuff is just linked to your phone! My phone randomly died a few weeks ago and I wasn't that worried about my account because I thought it was the one thing that would be connected to an account and I'd just be able to restore everything but it wasn't and the only thing that you can do in that situation (if you don't have access to the phone your account is on) is make a whole new account! Luckily, my boyfriend was able to get my original phone to turn on for 5 minutes so I was able to link it quickly (it's a really fast process!) before it died again but otherwise everything would've been gone. TT ~ TT Learn from me!
Happy for you that you managed to recover your save safely!
I have a similar story to share. In July my phone randomly and without any warning decided that it was time for its screen to die. It just went completely black. After a not so easy to do restart it turned greenish with some artifacts (no visible desktop or anything) for a short while, and then went black again. The touch function was still working but I had no idea what I was tapping, and there weren't any sounds either because my phone is always on silent.
So obviously my main concern was my Pocket Camp save. I wasn't worried about my photos or files, because I was sure I could recover them one way or another, but I had no idea how I would upload my PC save to the servers.
I used this one desktop app that allowed me to operate the connected phone through the PC. It was a little tricky to allow access blindly, but after many tries and errors I managed to learn where I needed to tap my finger. I was able to copy all the files and run most of the apps, but not every one worked correctly, for example bank and payment apps obviously restricted third party operations. I tried to upload the save to the cloud using this app, but it wouldn't work, there was an error every time. And so it was time for my second round of panicking.
Then I searched the Internet for any new solution and found out that the Pocket Camp Complete save is stored locally on the phone, and I could just copy it from the files. I'm pretty sure I knew that first part, but I hadn't connected the dots until then. And that was pretty much it, I copied the save files into another device and it worked. I was so relieved.
Since then I back up my save manually every few weeks. I know I could somehow set it up to get it done automatically, but I don't think it's that important. Even if I lost my phone save now, only a few weeks, or at worst month or two of progress would be gone, not everything. Sometime later I also learned some other save file related things, but I'm not sure if I can talk about them here.
tl;dr: The save is stored locally on your device and you can back up the save files yourself.