Blankscreen - Yes, the council can invoke the Fair Access Protocol to get the school to admit a pupil ahead of the waiting list, for example if a pupil moves into the area outside the normal admissions round and there are no schools with places available within a reasonable distance. Also a pupil with an EHCP naming the school leapfrogs the waiting list.
Charmatt - In England the fair process is that late applications are processed after on time applications. If they are received before offers are made and it is possible to allocate them places they should be offered places at the same time as everyone else. If they are not offered places at that time they must go via the waiting list (if there is one - if the school they want has spare places they can be offered a place immediately). The waiting list starts operating the moment offers are made. Late applicants cannot leapfrog the waiting list nor can operation of the waiting list be delayed until 4th May or any other random date.
It is true that a number of councils seem to have a deadline of 5th May for late applications. However, late applicants cannot legally leapfrog the waiting list (if there is one). They can, however, take spare places at any time. And this "deadline" is meaningless. The council cannot refuse to process an application just because it has been received after 5th May. I suspect that, since many councils want parents to accept or reject offers by 5th May, the "deadline" is simply to encourage late applicants to get their forms in before waiting lists start moving.