In the UK, you would find it hard to justify to a BRC or Tesco auditor if you had wooden pallets in a high risk / high care area; this is not only because of the foreign body risk but also because of the higher incidence of water in high risk / high care and the risk of mould growth (yuk).
Here is how I have controlled it in the past in high risk / care factories:
Ingredients / packaging etc can come in on wood, ideally good quality undamaged chep pallets (added benefit of being blue so contaminants may be spotted). There must be inspection at intake and either transfer to undamaged pallets (plus a complaint to the supplier) or rejection if the pallet is unsuitable (and I have used pallet inverters in the food industry but it's better to get your suppliers on board). There must be a layer card / plastic between the pallet and the product (especially if you're looking at produce) or else you risk splinters in the food.
Due to the process flow; the food or packaging will either have to be washed, sanitised or cooked into high risk so inevitably for that it will need to be depalletised. I always have specified deboxing areas (as card can also be a risk) and the wood doesn't go past there. If you're deboxing to cook with, the ingredients then go into trays or tote bins.
Wood is then not used again until despatch where there is already the manual handling process of packing and palletising anyway; the product is already wrapped at this point.
In the ambient food industry in a plant with no barriers
I always have tried to minimise the presence of wood near open food, this means to me only using wooden pallets for packed products or ingredients (wrapped / protected as before) and not using them for work in progress. The way I've used to simply acheive this in the past is to mark out where pallets are permitted on the floor.
I hope all that is helpful. I agree with the other posters that plastic pallets are not foolproof and bring their own risks.
A word of caution. In the food industry, I have seen 3 different freezers at different companies with wood hazards in. This is because freezers normally have racking and the lazy option if using a frozen ingredient is to take a small amount out of the freezer and leave the box open (or it falls open due to the lack of adhesion on the tape due to the temperature). It's normally the small use ingredients on the bottom rack, e.g. frozen herbs and more often than not, a damaged wooden pallet is on the rack above. We once had a wood find in a meal and that's what we traced it back to. In my view the only way to really get rid of that problem is to have no racking in freezers (plastic pallets cannot be used in freezers as they become brittle.)