Beta key are no longer needed to unlock the Garry’s Mod Beta. It’s available to everyone that owns Garry’s Mod. You should be able to click here to install it.
The switch over is due to happen on the 24th September, at which point Garry’s Mod 12 will cease to exist - and 13 will replace it. Due to the extensive engine changes it’s likely that all demos and saves will no longer work, or at best work only partially. So I’m urging you to start using 13 if you can.
Although the 24th September is the target date - I am considering extending that to some time later in October - depending on how it’s looking nearer the deadline.