Time Machine shows no backups - but there is data on the disk - how do I get at that data?

Hi all, got an issue I hope someone can help with.

I spilt liquid on my machine - sent away, fixed etc.

I've plugged in my Time Machine drive and using Migration Assistant tried to restore, but Migration Assistant continues to look for the drive without seemingly finding it. After 30 seconds or so the blue light on my SSD Time Machine Drive goes off, and the on screen wheel spins and spins.

If I log into my machine the drive does show up as Time Machine. However it shows 'no completed backups'. However, using 'get info' in the finder I can see there is 1.6tb on the Time Machine (which is exactly what I'd expect from the last backup). But I can't find a way to actually see any of these files.

If I try to use my SSD Time Machine on the newly restored Mac it says "Should this computer claim the existing backups on the disk “Timemachine”? So to my logic, it is recognising that there is a backup on the SSD Time Machine, but I don't seem to be able to find that backup through any mechanism that allows me to use it to restore.

So, why can I not see any backups although there is data there, and how might I get access to that data, either through a restore or even just being able to look at the files on the Time Machine and take what I need?