Definitely could come on quickly, or over time. (same for people -- usually what seems 'quickly' was actually over time til a tipping point was reached for the allergy). Are you working with a vet or specialist? He really needs a visit to the vets if not, as figuring out an allergy is often a very slow and careful (and often, tedious!) process -- not as simple usually as just stopping a single food. You would be most likely to have success by working closely with either your own vet or if this isn't an area they know a lot about, going to a specialist.
Unfortunately given the breed, and if you cannot find a specific reason for the scratching and paw chewing, I think you will need to also consider seeing someone with a good familiarity with syringomyelia as what you are describing would also be typical for dogs with SM. A lot of people waste months and years looking for causes of allergies when the problem is actually SM, and important time may then be lost, during which more neurological damage can be done. So it is important to for now, keep that possibility in the back of your mind as you seek some answers. Do not let a vet dismiss such concerns -- the board here has many people whose vets kept telling them the problem could not be SM. Exhaust all other options first, but realise you are describing something which in any case, whether allergies, a skin problem, or SM, really needs professional advice so you can carefully and methodically eliminate the possibilities and hopefully, find an answer.