OK, maybe the Rebellion gets word from Saw, he knows of a weapon and has details, but will only reveal it to Jyn for {reasons}. The Rebels break out or release Jyn from their own custody (hence the bit in the early trailers about being trusted without her shackles). Turns out he wants to reveal that her father is alive and that he knows where to find the plans. Meanwhile, the DS’s weapon test is on Jedha, which is successful, but they escape without seeing what caused the blast.

Not sure about the crashed X-Wing though.