FYI, One possible way to do what you want is:

while (<DATA>)
{
  next unless /^~/;# Skip line unless it starts with ~
  tr/_/+/;
}
continue
{
  print;
}
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The reason your /g version doesn't match is because /g doesn't try to match several times from the start of the string, but from the position last match. So after turning "~a_b_c_d" into "~a+b_c_d" it will start looking after the +, in the substring "b_c_d". Because this substring is not at the beginning of the string, ^ fails, (and ~ would obviously fail as well).

The \G anchor, meaning "from the last match" let's you express "and underscore after a ~ or any underscore after that":

  pos($_) = -1; 
  s/(^~ | \G) .*? \K  _  /+/gx;
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(^~|\G) can either match a ~ at the beginning of a string, or at the position of the last match. But since I have forced that position at -1, ^~ is the only possible alternative on the first try. Beyond that point, the s/// will run in a loop, where ^~ fails (because not at the beginning of the string), but \G matches the position of the last iteration.

NB: because $_ is the default variable, pos($_) = -1 can be simplified to pos = -1

NB2: and the reason jwkrahn's solution works is because instead of using /g, it calls the s/// operator in a loop, meaning it starts over again on each iteration (after turning "~a_b_c_d", it will run on "~a+b_c_d").