Would something like a log file work:
wget -nv -o wget.log -O "save-file" "URL"
And if you want to keep adding to the log file instead of overwriting it:
wget -nv -a wget.log -O "save-file" "URL"
What you get is something like:
2017-05-15 16:46:58 URL:https://some.url/ [file size] -> "save-file" 
What is happening here:
-nv : "not verbose" output. Just the details, not everything.
-a wget.log : "append" info to the log file (in this case, "wget.log").
-o wget.log : Creates a new log file (old one is removed, if found).
The rest you most likely know.