Success!
First and foremost, I am grateful to Bob (docmoore ) and Jorgen for having given me of their time and their advice. Thank you guys.:salute:
I did not use the Time Machine B/U.
I used the Carbon Copy Cloner and the clone I had made of my now sick iMac; on an external HD. ( I had actually made 3 clones!! ).
I tested my Mackbook Pro with one of the cloned HDs. Fired up the system while pressing <option> key, so as to choose the boot up disk.
I choose one of the connected clones.
I started the CCC app on the cloned disk. There was a notification that a new version of CCC was available and would I like to update.
I did. That was a mistake!! Either something in the cloned disk or incompatibility of the b/u with the previous version of CCC, I do not know.
I then went through the same routine but with the 2nd of the CCC clones. This time I did not upgrade to the latest version. I choose to restore only some of the files;
omitting all the photo image files, and other unnecessary clutter.
Went to have coffee at Starbucks! Honest.
On my return about an hour later, I was greeted with a message on the screen informing of a successful restore!!
But how to remove the external HD? I couldn't eject it as it was the start-up disk.
I just shut down my MacBook, and restarted with the <option > key pressed.
Choose the MacBook disk as the start-up.
Most relevant apps, inc. Adobe, asked for the license key. Have them handy.
Ejected my external HD, safely.
All apps were run successfully.
A special thanks to the folks at CCC, for having responded to my queries.
The CCC software is highly recommended. Its value is found when the HD or the system crashes.
It also maintains the correct hierarchy of all the files during the restore process.
Biggest issue was that getting the connectors sorted out...I was running usb-A/B ports.
MacBook Pro only has the usb-C ports....get your connectors sorted.
Have all passwords ready, cables ready, do not upgrade to any newer versions of sw until restore is completed.
And do have multiple b/u, clearly labeled as to the OS X version, and date of last b/u..in case of incremental b/u.
Hope this helps someone non Apple tech savvy soul somewhere.
I shall let you know how the iMac restore goes, if and when it is discharged from the ICU.
Best regards.