The London Evening News of 14 February 1931 has a notice about extra trains from Waterloo to reach Twickenham for that afternoon's England-Ireland match. This included "Special arrangements have also been made for taking the crowds home after the match. Fast trains to Waterloo will leave the new Twickenham station, and trains calling at intermediate stations will go from the old station."
There is a 25" OS map dated 1935/36 that shows the access road to the new station in place (more or less as now). The Up side sidings have gone, and between the signal box and the nearest track is an outline that could well be a platform. That track runs round the outer side of the Up platform at the stations each side. I assume the platform was a simple, bare, improvised one.
So it looks as if some limited work and use of the site had started, though perhaps not as part of the new station design - which may not have existed then.