Living Images on the Nokia Lumia 1020 has been disabled (for now) because of stability issues.
Living Images works by creating a short mp4 video of the last few frames that the camera sensor sees before the shot is actually taken: It reads from the full sensor image (20-MP in the case of the 1520/930/Icon) and converts it into a 1080p video (the screen resolution of all three phones). This task is highly dependent on three factors: the phone's RAM size, processor speed, and camera sensor resolution. Living Images works by default on the 930 because it's a feature that's designed to work specifically for the 930 hardware: 2-MB RAM, quad-core 2.2-GHz processor, and 20-MP camera. It works on the Icon because the Icon is basically a rebranded 930, and it works on the 1520 because the 930 and 1520 share almost the exact same hardware except for screen size.
Now, as for the other Lumia models, it works on the 520 because even though the 520 just has a dual-core processor and smaller RAM, it also has a lower resolution camera, one that doesn't do the supersampling of the 1520, 930, and Icon. This means that the 520's processor and RAM have no problems making short videos out of the lower-resolution camera.
The 1020, however, poses a unique problem. While it does have 2-GB RAM, it only has a dual-core 1.5-GHz processor that has to deal with a 41-MP sensor (actually 34 or 38, depending on your aspect ratio). That's TWICE the camera resolution of the 930/1520 at HALF of the processing capability. It was quickly found out through Lumia 1020s running the Nokia Camera beta application that the processor gets bottlenecked because it ends up doing three things at once: Creating a full-resolution 34MP/38MP photo, creating a 5MP supersampled photo, and creating a Living Image video (previously, it only had to do the first two things). This results in either the Living Image not being created at all, or the Living Image appears, but the native high-resolution reframe-able image is not created. In some cases, the shot isn't processed at all.
THAT is the reason why Living Images isn't supported on the Lumia 1020 yet. I say "yet" because there's a good chance it may still come to other Lumia devices. If any of you remember, the Nokia Camera app was originally exclusive to the 1020, then later to other units with PureView cameras such as the 920, then eventually to all Lumia devices as the app got more optimized. It's likely that the same will be done to Living Images, particularly, when Nokia/Microsoft streamlines the feature to work on dual-core hardware (and to compensate for the 1020's huge sensor).
This is the same reason why Continuous Autofocus is not yet available on other devices either.
My advice? Just wait it out. It will come eventually. Besides, if you lived with your 1020 without Living Images until now, chances are, you can continue living without it.
As the owner of a Lumia 1520 and a Lumia 1020 (both of which are now running on Cyan), I can tell you that while Living Images is nice to see on my 1520, it's hardly essential and definitely not worth sacrificing the 1020's exemplary 34/38-MP reframe-able photos. Continuous Autofocus would definitely help the 1020 reduce shot-to-shot times though, and so I would rather that Nokia/Microsoft prioritized that over Living Images.
I hope that clarifies the situation for a lot of you.