Most probably you're now using /etc/bash_completion – check if it's loaded by your profile or bashrc.
It contains a function _expand which replaces the tilde by the path “to avoid problems when $cur starting with a tilde is fed to commands and ending up quoted instead of expanded” (from the comments in the function).
If you still want to keep the tilde, you could have a look at the _filedir function. Close to the top there's the line
_expand || return 0
Remove it or disable it with a # prefix. This is a ugly hack, however, and you might run into the problems mentioned above.
