To build on Apiman's answer, it's more likely in general you'll find the PATH in HKLM\SYSTEM\CurrentControlSet\Control\Session Manager\Environment
instead, which contains the system PATH instead of the User's PATH. I've also made a few corrections below.
Run this in the cygwin environment to load the Windows system PATH (or other environment variables by changing var_name)
export var_name="PATH"
export $var_name="$(cygpath -pu "`reg query 'HKLM\SYSTEM\CurrentControlSet\Control\Session Manager\Environment' /v $var_name | grep $var_name | cut -c23-`")"
Of course with the, code above, the windows PATH will replace the local PATH, making you lose access to cygwin /bin and others. Instead, you probably want to append the Windows PATH to the cygwin PATH:
export PATH="$PATH:$(cygpath -pu "`reg query 'HKLM\SYSTEM\CurrentControlSet\Control\Session Manager\Environment' /v PATH| grep PATH | cut -c23-`")"