Collections are like virtual folders, stored in your catalog. There is no .plist file or anything else in your Application Support folder related to collections. You add images to a collection by simply dropping them into the collection, and that is the problem: there is no info anywhere that stores in which collection an image used to be but is no longer. That means there is no way to restore this unless you have a catalog backup. If not, you will have to add the images again manually.
If you still have the old computer and your original catalog is still on it just Copy the catalog file over to the new computer using something like a USB thumb drive.
When copying the catalog to the new computer put it in the same folder as the one that got created with Time Machine and when asked if you want to Overwrite the other file just click yes.
There have been reports of problem when using Time Machine to restore LR program and catalog files to a new computer.
Actually, I did just that...copied the catalogue over and then pointed LR to it. It has taken 3 days but my Collections are slowly appearing.
But to be safe, I went back two backup catalogues three days ago and that is the one I am pointing to. It shows that my iMac>LightroomPhotos folder has 32,000 images, but my Catalogue has 64,000 ( I think because when I pointed to it I was asked to "Add" to the catalogue.
My old external HD is showing up listed in Folders but has a "?" next to it because it is not connected.
My question is what to do with the "duplicated" images noted in the Catalogue while the actual pictures are the same 32,000.
I am worried that maybe this last "Add" function is why my Collections are now correct (not duplicated). So before I mess around with deleting anything from the Catalogue to prune back to 32,000 I wonder if you have any advice?
Sounds like you imported your 32,000 images again instead of 'reconnecting' Lightroom. That is the wrong approach, because that creates duplicates without any edits or added metadata. Start again from a backup catalog, and this time use the following procedure:
Do not use Import or Synchronize Folder! Right-click on the top 'missing' folder (folder with question mark) and choose 'Find Missing Folder'. You will get a standard 'Open' dialog. Navigate to the current location of that folder with this dialog and select it. Done. That will 'reconnect' the database to the folder and all its subfolders. If you don't have one single top folder, you'll have to repeat this for each of them.