For $d<4$ the Fourier transformFourier transform with respect to the radial coordinate $y$ has a closed form expression in terms of a Bessel function:
$$\hat{G}(\xi,t)=(2\pi)^{d/2}\xi^{1-d/2}\int_0^\infty H(y,t)J_{d/2-1}(\xi y)y^{d/2}\,dy$$ $$\qquad=\frac{1}{\Gamma \left(\frac{1}{2}+\frac{1}{d}\right)}2^{\frac{d^2+d-2}{2 d}} \pi ^{d/2} (t/\xi)^{\frac{d-1}{2}} (t \xi)^{1/d} K_{-\frac{d}{2}+\frac{1}{2}+\frac{1}{d}}(t \xi).$$
This can then be Fourier transformed with respect to $t$, in terms of a hypergeometric function,
$$\hat{H}(\xi,\tau)=2\int_0^\infty \cos(t\tau)\hat{G}(\xi,t)\,dt=$$ $$\qquad = 2^{d} \pi ^{d/2} \xi^{-d} \Gamma \left(\frac{d}{2}\right) \, _2F_1\left(\frac{1}{2}+\frac{1}{d},\frac{d}{2};\frac{1}{2};-\frac{\tau^2}{\xi^2}\right).$$