We present a single walled carbon nanotubes (SWCNTs)-coated tilted fiber Bragg grating (TFBG) hot-wire anemometer (HWA) with simple configuration, linear response, and high sensitivity. TFBG is utilized to effectively couple a pumping laser at 1550 nm to the cladding mode that is absorbed by the SWCNTs film immobilized on the fiber surface with good light-heat conversion efficiency. As a result, the TFBG is converted to a "hot wire", and the wind speed can be deduced from the output power of the laser, which is a function of both the wind-induced temperature change and the spectral profile of the cladding mode. The most significant aspect of the HWA system is that we use the Gaussian shape of the high-order TFBG cladding mode to compensate for the inherent nonlinear relationship between the heat loss and the wind speed that is an undesirable characteristic of existing HWA systems. The validity of this novel operating principle was verified theoretically and experimentally. Via careful control of the parameters, a good linear response of the HWA system was achieved, especially for the low wind speed range where nonlinearity was more conspicuous. It was demonstrated that, with a low input power of only 29.3 mW of the pump laser, an R2 value of 0.9927 was obtained in this fiber-optic HWA system with high sensitivity 7.425 dBm / (m/s) and resolution 0.0027 m/s in a small wind speed range (0-2m/s) considering the intensity resolution of OSA and the noise of the pump laser. Furthermore, the system also exhibits a simple and low-cost design with only one laser source and one low-cost power measurement component.