In this paper, a rateless coding scheme with enhanced unequal error protection (UEP) property is proposed by concatenating the systematic polar code (SPC) and spinal code, where SPC is used as the outer code and spinal code is the inner code. The novel concatenated polar-spinal coding scheme is designed to provide an improved UEP for the information with different reliability requirements. At the receiver, a tail CRC-aided Bubble decoding is proposed to improve the performance of spinal codes effectively. Moreover, the rateless transmission control is designed carefully for the time-varying channels. The total complexity of joint concatenated decoding process is reduced. Simulation and analysis verify that the novel rateless polar-spinal coding scheme can provide better UEP performance for the information of each importance level in contrast to some existing UEP rateless coding schemes.