Patrick McLaughlin correctly guessed that the Council of Liberal Churches was the precursor to the UUA. In 1953 a Joint Unitarian and Universalist Assembly was held that authorized the creation of the CLC. Since it was legally impossible for the AUA and UCA to merge (as previously explained), the Council of Liberal Churches was an entirely new entity funded by the Unitarians and Universalists. The idea was to slowly transfer all of the assets to the CLC and then close the AUA and UCA, effectively bringing about a merger without ever actually merging. This plan got started in 1953 when the AUA and UCA closed their educational departments and merged them together under the umbrella of the Council of Liberal Churches. In 1956 the Publicity and Promotions departments were shifted to the CLC.
The Council of Liberal Churches did not satisfy everyone, and ultimately the plan did not come to full fruition–instead, a plan for consolidation was hit upon and that is how the Unitarian Universalist Association was created. However, had the legal loophole of consolidation not been discovered, the CLC is probably what would have come into being as the eventual full merger of the Unitarians and Universalists. That would mean that there would be no UUA and in fact no Unitarian-Universalists or Unitarian-Universalism: instead, we’d all be known as Liberal Christians, Liberalists, Liberals, or some similar moniker, and our actual faith would be known as Liberal Christianity or perhaps Liberal Religion. Although the effect would’ve been about the same, it is strange to consider what it might have been like with a very different name for the denomination.
By the way, Patrick mentioned 1825 in his comment. This is probably just a slight confusion: the American Unitarian Association was created in 1825, whereas the Council of Liberal Churches came on the scene 128 years later.