I know that closed loop hall effect sensors generate a current in the secondary winding to oppose the flux in the core such that the net flux will equal zero. At this point the current in the primary is proportional to current in the secondary winding, however when the flux=0 will the hall effect sensor simply not switch off? Then secondary winding current will start to fall off therefore causing the hall sensor to turn on again i.e the output would be oscillatory?
The Hall sensor used for this type of system is not a switch type (with a built in comparator) but a linear output, with a voltage that's proportional to flux.
The amplifiers in linear Hall sensors do have a finite loop bandwidth, and the secondary winding drive will certainly be slow(ish), so it's possible or even likely to get an unstable loop if you simply wrap it round an op-amp without thinking about stability. Like all high order systems, it needs careful design to make sure the unity gain bandwidth drops through unity with a phase shift of less than 180 degrees.