You had one system that contained a number of video files, then a new OS installation was performed.
If the new system was installed without reformatting, then the files may just be there, not needing recovery at all. (Check what folders are in
/home; perhaps an old home directory from a previous installation is in there, next to your own home directory in the new installation.)
However, if the partitions containing the video files were removed or reformatted, and then data was written on the disk where they existed, it's highly probable the new data (from the newly installed operating system) irreversibly overwrote large parts of the video files. It's likely some parts of all of them were overwritten. This is hard to recover, it could take hours, days, weeks, or you might not succeed, and it would involve trying to repair damaged video files, since it's unlikely you'd be able to recover the files intact.
Unless the TV episodes you lost are rare and cannot be obtained again, it's not worth it to attempt recovery. What you'll likely get from recovery after the drive was formatted and another OS installed is, at best, highly corrupted files. While better outcomes are possible, probably most of the files would cut out and stop playing altogether, somewhere in the middle. It would be one thing if you had just accidentally formatted it, but once a overwriting has happened, data recovery is only worthwhile in cases where valuable, difficult-to-reconstruct data are lost.
It's a hassle to get the files again, and takes time, but so does data recovery, and data recovery will likely take more of your personal time and effort (while preventing that computer, or at least that hard drive, from being put to productive use).
Usually, I only recommend attempting data recovery if the likelihood the data has been overwritten is very low (not the case here), if the value of the lost data is very high (probably not the case), or if the data has intangible value that's hard to quantify but you are very attached to it (e.g., your own creative/artistic works regardless of whether or not they are monetizable; the only copies of pictures of your children or from a vacation with a now-deceased relative).
Still, if you want to try, there's a slight possibility you may recover significant data. It's possible that, by chance, the files were mostly located somewhere on the disk that was not overwritten by the new OS installation.
If you want to attempt recovery, first take a look at this page:
You would image the drive, then perform analysis on the image to recover files from it. Photorec (it's not just for pictures) might be a good analysis tool to start with.
Also I recommend taking a look at this question:
Some of these other questions may also be of help.
mkfs.ext4) only? Did you install the new operating system already before you noticed? It is important to know the level of data recovery you need to recommend software for it.