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My friend dropped tea on her laptop, hence there was no data seen on the disk.
She took it to a laptop repair company, who said the disk was burnt out and the data had been lost.
She has some very important stuff on there, so she asked me to have a look, as I have used TestDisk before to recover data successfully.
Firstly, I tried Windows Explorer, to see if Windows could see it, with (as I thought) no success.
I then connected it to Ubuntu via my USB SATA caddy and it would recognise the disk in Disk Utility but not in the 'Files' app under Other Locations. In Disk Utility, the SMART data is fine, the disk is ok with no bad sectors.
I then recovered some files successfully (I tested some by opening them.) However, the disk is 1TB and the SSD in my laptop is only 256GB so it filled up pretty quickly (and also because I don't have permission, Ubuntu won't let me delete the files!)
Anyway, after I tested recovery with some files onto my SSD, I decided I wanted to put them straight on to my 4GB Seagate External Drive. I plugged the drive in, it was found by Disk Utility, Ubuntu, and it was included in the list of disks when TestDisk initially starts.I made a note of the mount point (/dev/sdb in this instance) and set about to select the files I wanted to recover. (My Seagate external drive has got two partitions, one which is only 1.34MB, and the rest, which are /dev/sdb1 and /dev/sdb2.)
After the file selection, I pressed a to select all the files in a folder, and then told TestDisk where to save the data. I selected .. then .. then dev and sdb2. It said it copied all the files ok with no failures. I then checked on the disk in Files, nothing there. My external drive was not listed in the media folder either.
If it found the drive initially, why won't it copy to it?

If Testdisk cannot see your external HDD as your title suggests, please consult with a professional recovery lab.
Otherwise please change the subject of your posting in a way that better describes your problem to receive the desired answer to your question.

Although I use Linux rather rarely I think I can tell you what happened according to your description:

When you are copying files from one location to another, both locations have to be somewhere in the file tree. That applies to Linux as well as to Windows.

You selected your source files correctly but you did not specify a file location like a folder for instance.
Instead you selected a raw device.
Now Testdisk certainly copied your files to that device - right to the beginning where the MBR or the GPT is located and the first partition usually starts. That is outside of any file system located on one of your partitions.

I guess all your files are there now - unfortunately each one overwriting the preceding one because the write part of each single copy operation started at the beginning of the device.

You probably logically dammaged your target drive now.