Industrial and construction wastes make up about half of all world wastes. In order to reduce their negative impact on the environment, it is possible to use part of them for concrete production. Using experimental–statistical modeling techniques, the combined effect of brick powder, recycling sand, and alkaline activator on fresh and hardened properties of self-compacting concrete for the production of textile-reinforced concrete was investigated. Experimental data on flowability, passing ability, spreading speed, segregation resistance, air content, and density of fresh mixtures were obtained. The standard passing ability tests were modified using a textile mesh to maximize the approximation to the real conditions of textile concrete production. To determine the dynamics of concrete strength development, compression and flexural tests at the ages of 1, 3, 7, and 28 days and splitting tensile strength tests of 28 days were conducted. The preparation technology of the investigated modified mixtures depending on the composition is presented. The resulting mathematical models allow for the optimization of concrete compositions for partial replacement of slag cement with brick powder (up to 30%), and natural sand with recycled sand (up to 100%) with the addition of an alkaline activator in the range of 0.5–1% of the cement content. This allows us to obtain sustainable, alkali-activated high-strength self-compacting recycling concrete, which significantly reduces the negative impact on the environment and promotes the development of a circular economy in the construction industry.