The question's premise is wrong.
River does not end up in Alliance hands between the series and movie.
The scene you are referring to is a holographic recording.
You can tell it's a hologram because of the lighting difference between the room and the image being projected.
It eventually details how Simon rescues his sister (watched by 'The Operative'), which leads you up to the beginning of the Firefly series, and cuts out to the beginning of the movie.
The script also backs this up,
THE OPERATIVE (O.S.)
Stop.
The action freezes.
(CONTINUING)
Lovely. Lovely. Backtrack.
The action REVERSES, taking us back to the moment of Simon and River on the gurney just before it rises.
THE OPERATIVE (O.S.)
(CONTINUING)
Stop.
There is a motionless beat, River frozen in that crouch, and he steps through what we now see is a hologram of the event.
And then later on:
THE OPERATIVE
River was your greatest success. A prodigy -- A phenomenon. Until her brother walked in eight months ago and took her from you.
(emphasis added)
So not only are the events a hologram, but the script shows they occurred 8 months prior to the film, which could easily place them before the series began.