Siliceous fly ash (FA) is the main additive to currently produced concretes. The utilization of this industrial waste carries an evident pro-ecological factor. In addition, such actions have a positive effect on the structure and mechanical parameters of mature concrete. Unfortunately, the problem of using FA as a Portland cement replacement is that it significantly reduces the performance of concretes in the early stages of their curing. This limits the possibility of using this type of concrete, e.g., in prefabrication, where it is required to obtain high-strength composites after short periods of curing. In order to minimize these negative effects, this research was undertaken to increase the early strength of concretes with FA through the application of a specifically formulated chemical nano-admixture (NA) in the form of seeds of the C-S-H phase. The NA was used to accelerate the strength growth in concretes. Therefore, this paper presents results of tests of modified concretes both with the addition of FA and with innovative NA. The analyses were carried out based on the results of the macroscopic and microstructural tests in five time periods, i.e., after 4, 8, 12, 24 and 72 h. The results of tests carried out with the use of NA clearly indicate the possibility of using FA in a wide range of management areas in sustainable concrete prefabrication.