During avian morphogenesis, feather follicles generally emerge row by row, in a medio-lateral sequential pattern through a two-stage developmental process. Initially, dermal densification occurs to render the tissue competent. Subsequently, a complex interplay of signaling between dermal cells and proteins leads to the formation of dermal cell aggregates, which initiate feather buds. This study employs a reaction-diffusion-chemotaxis mathematical model to analyze the speeds of these stages and their interactions, to determine the overall emergence speed of feather follicles. Our model enables us to calculate the speed of the competence wave and, through a weakly nonlinear analysis, approximates the emergence speed of follicles within a competence zone. Numerical simulations suggest that when the emergence speed of follicles surpasses the speed of the competence wave, the bud emergence becomes synchronized to the competence wave's speed. Conversely, the formation of follicles does not affect the competence wave. This theoretical and numerical work provides a precise approximation of bud emergence speed in initially naive tissue, highlighting the complex dynamics of feather follicle morphogenesis and the significance of understanding the interplay between dermal competence and cellular aggregation in this process.