You can lose your whole digital world with just one slip-up: when you format the backup drive instead of the new one, wipe the memory card in-camera before importing images, or the Windows installer deletes the entire contents of the drive that held everything. A quick format is loud but shallow; shallow is recoverable, as long as nothing is written over it.
A quick format (this has become standard across most platforms) creates a completely blank file system: a brand new index at the beginning of the drive, with all of your original data left intact behind it. As recovery scans past the now clean area at the start of the drive, it discovers the remaining structure or cuts out each file individually from the mass of data and usually recovers the bulk of what existed on the drive. However, a full format on newer versions of Windows is much worse (it zeroes the drive as it goes), and we feel obligated to be honest about this: once you have completed an erase, those areas of the drive are completely lost. If it was interrupted, the diagnostic will tell you exactly how far into the process the zeroing was going before it stopped. The single best thing to do regardless of which method you used for erasure: don’t put anything else on the drive, and if the format was step one of reinstalling Windows, do not go on to step two on that disk.
On the bench, we see the exact same four stories repeated over and over again: the two-drive mix-up (drive letter "D" intended for one of the drives but clicking on drive "C"); the OS reinstall that formatted the data partition instead of skipping over it; the 'initialize disk' prompt you answered when the partition table for an older hard drive did a little jump, effectively erasing a drive that was just disoriented; and the in-camera format that happened before you had even imported your photos into the computer. Each of these will produce different signatures and recover differently as well (which folders, what type of files, and whether they have their names associated with them), and the free diagnostic will tell you exactly which of those stories your drive is going to be about before you spend a penny.
If this was a quick format, and the disk just sat still after that: generally, yes, for the most part. The new empty index is simply sitting at the beginning of your prior data (like fresh paint on a fully filled filing cabinet). The odds decay solely due to new writes, so all the drive has to do now is remain powered down.
Harder, honestly: the install process formats and writes gigabytes of OS to the top of your hard drive. Everything else that was on there (which would be most of the space in a larger HD) can still be recovered; you might not know what some of these files were named, but they're often still intact. The diagnostic also provides a map of how much has been overwritten and will list what survived at no cost.
Each new shot may have landed on an old shot, but because cards are large and shoots are relatively small, you'll probably end up with a partial recovery, with the best chance being what wasn't touched by the last session. Stop shooting on the card now, and the diagnostic will count what's left.