The story is at least semi-plausible, but Turing also still had friends in the right places (not enough to dig him out of the hole he got himself into with the local plod*) and there was a strong social taboo around suicide.

(* At the time there was good reason to believe that the end of the outlawing of homosexuality was just around the corner, so offering a genuine explanation was not necessarily Turing acting as such a naif as is often portrayed.)