This is not possible with a regular Find and Replace. If you use Notepad++ 6, you can take advantage of the new regex engine that supports PCRE (source).
Press Ctrl + H to open the Find and Replace dialog and perform the following action:
Find what: \^.*?~
Wrap around: checked
Regular expression: selected
. matches newline: checked
Now press Alt + A to replace all occurrences.
The regular expression in Find what is composed as follows:
\^ is a literal ^.
.*? is the least amount of characters that allows the regular expression to match.
~ is a literal ~.