If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is:

The speed of a transverse wave on a stretched string is $v=\sqrt{\frac{T}{\mu }}$, where $T$ is tension and $\mu$ is linear mass density.Let initial tension be $T_i=T$ and final tension be $T_f=2T$.
$$\frac{v_i}{v_f}=\sqrt{\frac{T_i}{T_f}}=\sqrt{\frac{T}{2T}}=\frac{1}{\sqrt{2}}$$
Thus, the ratio is $1:\sqrt{2}$.