When I read the title I assumed we were going to deal with deleted data recovery. This isn't a big issue at all, your data is safe, so don't worry. As for how we go about recovering your data... the most safest way would to transfer the drives into another computer. If you have a laptop then you can use one of those SATA to USB adapters. If you don't have either a computer/laptop to plug in those drives, I suggest leaving them as is, until you get one. Again, your storage media is not compromised, so your data is safe.

Now onto your current computer problem. That was quite the chain of events lmao. I have two theories here, either you picked the wrong GPU setting in your BIOS or you picked the correct one but still have your monitor connected to the GPU's HDMI slot instead of the Motherboard's. If it's the former, just swap the cable and you should be good If it's the latter don't worry, it's an easy fix. Open up your side panel and remove the little round CMOS battery from the motherboard. Wait 10-15 minutes and plug the battery back in. Your BIOS should reset and default to your Nvidia GPU.