In recent years, the derivation of nonasymptotic converse and achievability bounds on the maximum coding rate as a function of the error probability and blocklength has gained attention in the information theory literature. While these bounds are accurate for many scenarios of interest, they need to be evaluated numerically for most wireless channels of practical interest, and their evaluation is computationally demanding. This paper presents saddlepoint approximations of state-of-the-art converse and achievability bounds for noncoherent, single-antenna, Rayleigh block-fading channels. These approximations can be calculated efficiently and are shown to be accurate for SNR values as small as 0 dB and blocklengths of 168 channel uses or more.