Here's my attempt which seems a little simpler than some of the others.

m[^ \d (?: (?<! \x20 ) \d | \x20 ){19} $]x
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Which says that the entire string must constist of a digit followed by 19 ((digits not preceeded by spaces) or spaces).

print m[^ \d (?: (?<!\x20) \d | \x20 ){19} $]x 
    ? 'Yes:' . $_ 
    : ' No:' . $_ 
    for @t;

 No:     123451234512345
 No:  123451234512345
Yes:123451234512345
Yes:1234512345
 No:123 451 2345
 No:
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