This is a really common situation that I know a lot of you are dealing with, and is especially frustrating if you are newer to the industry, which may mean that this rule that many studios have is preventing you from including your own professional experience on your demo reel.
My advice would be to ask your supervisors or the studio if you can use the footage. Many studios will simply not allow this, but others will be willing to compromise and let you use unrendered footage, or possibly rendered footage without the audio track. Of course, neither is ideal, but either is certainly better than nothing.
I know that some people won't include "non-allowed" footage on the demo reels that they mail out to companies, or their online reels, but have a special top-secret version of their reel that they bring to interviews to show in person. These reels contain the footage they aren't technically allowed to share, and the idea is that they don't leave that reel with the studio - only show it in one-on-one situations.
I've heard of this, but would be hesitant to recommend it, as it could potentially get you into trouble. There's a chance that the studio you are interviewing with will be unimpressed by your lack of loyalty to your previous studio's requests, and if they get the impression that you would do the same to them, it may hurt your chances of getting that job.
Most of the studios out there are more than familiar with this dilemma, and usually there is no solution but to wait for your movie to come out on DVD, the show to premiere on TV, or the game to be released. At that point, most studios allow you to use your footage on your demo reel.
In the meantime, your best bet is to be honest about your contribution to your previous project, provide references whenever possible who may vouch for your work, and if you feel like your demo reel isn't fully representing your abilities, then assign yourself a shot and animate it at home, and show off your most up-to-date best efforts!
Hope that helps. Sorry there isn't a good solution for this...