@Anon888 The answer is both yes and no. The overall official answer is that BAT on an unverified wallet is stored on the device. If that wallet is lost or damaged, you’re pretty much out of luck.
That said, @chriscat , Rewards Product Manager, had shared some information on how you can manually backup & restore files when moving to a new device. I’ll leave links to where he discussed it on Reddit below and will leave you to read it in detail. If you can get yourself to a scenario where you can follow the instructions, then it’s possible to get it all back. It just comes down to you being able to do each step “properly.” Make sure to read both links!
Keep in mind, it’s something they figured out we can do and offer it as a solution UNTIL they can do the official Backup & Restore. It’s not official or guaranteed. Just needing to stress that part. If you had done a verified wallet, which you haven’t, then just signing back in and verifying the wallet on a new device would have been all it takes.