As an important part of optical sources, nanolasers have a prominent influence in photonic circuit integration, and their nonlinear dynamics has become one of the research hotspots in recent years. In this work, we investigate the time-delay signature and bandwidth characteristics in a cascade-coupled nanolaser system, in which the master nanolaser is connected to an external feedback cavity and injected into the intermediate nanolaser and the slave nanolaser sequentially. The 0-1 chaos test is introduced to quantify the dynamics of the nanolaser, which can accurately distinguish whether the laser is in a chaotic state, and the autocorrelation function is used to analyze the time-delay characteristics in the laser output signal. This type of calculation has the advantages of fast operation speed, high accuracy and anti-noise robustness. The lower the autocorrelation value, the more difficult it is to extract useful information from the chaotic random sequence. The bandwidth is defined as a value where the range between DC and frequency contains 80% of the spectral power, a value that is only applicable to chaotic states. In the simulation, we compare and analyze the two cases of whether the intermediate nanolaser has a peak with obvious time-delay signature. The research results show that by selecting appropriate system parameters, the slave nanolaser can always output a broadband chaotic signal without obvious time-delay signature. Under the condition of a certain injection intensity, by changing the frequency detuning parameter, the intermediate nanolaser has an obvious time-delay signature, and then the slave nano-laser can output chaotic signals which can suppress time-delay signature and enhance bandwidth in a small parameter interval. When the time-delay signal of the intermediate nanolaser is completely hidden, the slave nanolaser can achieve the suppression of the time-delay signature in a larger parameter plane, meanwhile the bandwidth is significantly enhanced. In addition, by plotting the two-dimensional spatial distribution diagram and bandwidth line diagram of the output from the nanolaser under frequency detuning and injection intensity, it is determined that the nanolaser can simultaneously suppress the delay characteristics and enhance the bandwidth under chaotic signals. This provides an important theoretical basis for realizing the practical applications in secrecy-enhanced chaotic optical communication.