Cannot access TimeMachine backups from before clean install of macOS

Recently I did a clean install of macOS Catalina on my 2014 13" MacBook Pro. Before doing the clean install I backed up my computer with TimeMachine, and in fact I have been backing up with TimeMachine for years. After reinstalling macOS Catalina, my plan was to just go into the TimeMachine backups and pick and choose the files that I needed, since the vast majority of what I need day to day was already in iCloud Drive.

However, now when I into the TimeMachine Backups.backupdb folder, I see backups dated from before I did the clean reinstall but they have the name that I changed my computer to after the clean reinstall and they do not show any of the data from before the reinstall. In fact they just contain Macintosh HD - Data, Macintosh HD and Recovery folders, with Users inside of Macintosh HD - Data, with a folder for my current username (and not the username from before the reinstall) inside of Users, and that home folder only contains Movies, Music and Pictures.

In short, I can't access any of the data from before the clean reinstall even though TimeMachine has folders for all the backups going back over 6 months, but they all have the same structure as I described above and do not contain any of the data I need. The TimeMachine disk however shows just about the same amount of space occupied as before the reinstall, ~250Gb.

Is there any way that I can access my old files?

Time Machine can be tricky and sometimes you really need to reformat your external drive and start over. But you could check several things, like are you using HFS+ on your external Hard Drive? Did you repair permissions of your external hard drive with 'Disk Utility'? Is the internal and external hard drive name still the same? Is your user short name the same? You could also try, within the information panel of your ext. hard drive, to ignore permissions and user.