Access to time machine backups on

I teach digital media and have a lab full of 2017-2020 iMacs. One of my older Macs needs a repair, so the two students who use it will need to be on another computer temporarily. (At least, that's what we hope.) We back up all the computers to individual external hard drives using Time Machine. My question is this:

when we take the ailing computer in for service, how can my students get access to the content on their user ID stored on the hard drive when they are using a different computer. (I don't want to migrate all of it over. This isn't a new computer that they are using; it's just a temporary fix.) I know there is a work-around. I just don't know what it is.