Mixture models provide a logical approach to modeling complex data. The use of Lindley distribution to model different types of lifetime variables has made it attractive for further studies. The discovery of a fractional mixture of the Lindley distribution by time scale theory, and applications of the obtained distribution can be an exciting discussion. This work presents two fractional extensions of the mixed distribution that have applications in count data. For both parent and mixture distributions, the nabla and delta types of fractional distributions are considered: mixed nabla Poisson–Lindley distribution, referring to the situation where applied time scale for both parent and mixture distributions are nabla; and mixed delta Poisson–Lindley distribution, where both parent and mixture distributions are obtained using delta calculus. The parent distributions are special cases of nabla and delta power series distributions defined in this paper, and we call them nabla and delta Poisson distributions, and the mixture distributions are the nabla and delta fractional Lindley distributions. Some mathematical properties are obtained for nabla Poisson and nabla fractional Lindley distributions. Some estimation methods for one of model the parameters are studied. Extensive simulation studies are revealed to observe the performances of the estimation methods. Finally, a practical data example is carried out and it was concluded that the best model was nabla Poisson–Lindley distribution.