Once you've got the image in Ps and you've completed your editing simply closing the document will save it back into Lightroom. Back in Lightroom you should see the image with a stacked icon in the upper right corner of the thumbnail. One will be the .tif and the other will be the original. If you right-click on and Unstack the image you'll see the .tif beside the original. Unless you specified a location for local storage the .tif will be in the cloud as your originals are. To confirm that you can go to the browser version and confirm that you see that edited tif there.
However the file has not shown up in Lightroom, nor in my Photoshop recently opened files, nor anywhere on my computer.
The .tif edit file is completely missing.
Did you look in the folder on your drive? Maybe the file actually saved but didn’t get imported in Lightroom because you tried to quit to quickly. The tiff file would have saved to the same folder where the original raw is located. If it isn’t in that folder, then it didn’t finish saving.