Considering that the first X-1 airframe was rolled out in December 1945 it would appear to be difficult to believe that any input from the Miles project would have influenced the design. The movable horizontal stabilizer was always a feature of the X-1 but was for trimming purposes not primary flight control. Electrically operated it had a range of 5 deg up to 10 deg down. Conventional unboosted elevators were fitted.
The straight wing was there as they didn't want to introduce any more unknowns (like a swept wing) into the aircraft that they had to. An extremely thin straight wing has no problem going supersonic as proved by the F-104.