Your list is local to the browser in which you made it. To keep a copy, open Watchlist, choose Create backup and copy the complete text from the box into a note or document you control.
Restore without starting over
In another browser, open the same Watchlist page, paste the complete text into Restore backup and choose Merge backup. The tool adds valid saved titles and checkpoints. Existing checkpoints in that browser are kept rather than silently replaced.
Read the result
The confirmation tells you how many entries were added and whether any were skipped. Unknown titles, malformed data or episode numbers outside the recorded season cannot become valid checkpoints just by being pasted into a backup.
What the copy contains
It contains references and reminder values, not movies, passwords or viewing-service accounts. It does not transfer playback progress from another app. Avoid editing pieces of the backup unless you understand its format; keeping the whole text intact is the simplest approach.