We moved last year (mid school year) and found that the procedures vary a little from council to council. With some, you apply to the schools themselves, others, you apply to the council.
Oxfordshire council was happy to take our application with a letter from our rental agent, confirming that we had a lease in the catchment area, and with a copy of my husband's new contract, confirming that we were relocating to the area.
I'd suggest contacting the local council's education dept and asking if they'd accept an application with what you have.