It's the least easy to implement, but I have to say I agree with Glenn's suggestion.
In terms of the "allergens always on lower racks" approach, this is obviously good practice, but it sounds like the chiller may be multiple purpose in terms of storing both raw materials and (semi-)finished products? May also therefore need to think about best practice for "cooked"/raw segregation, which will add further complexity, although this may be less applicable depending on what constitutes "raw" (or semi-finished) for your process.
If full physical segregation isn't possible then some defined division of storage space will probably be essential.
Would I be correct in assuming that your products are open/unsealed at this stage? That certainly increase potential risk level, so if they can't have their own physically segregated area then I'd be looking at what can be done to accomplish this on a smaller scale.
The other point that would make me nervous in your position is the potential for people cutting corners to save themselves time. Do you have a rule about e.g. hand washing / PPE change between handling product with/without nuts? It could be quite difficult to monitor/enforce this if people are easily able to put down a tray of nut product and pick up a tray of non-nut product from adjacent racking.
Obviously there is always an extent to which we have to rely on our colleagues, but I'm generally of the view that wherever possible critical tasks should be as simple as possible and designed to make it harder to cut corners than to do it properly.
In any case I'd put in plenty of extra time on allergen training - it is gradually getting a bit easier as some incidents have had fairly significant coverage in the news/media, schools in the UK seem to be very much more attentive to such things etc, and thus there is more general awareness, but it can't hurt to reinforce best practice and definitely to make sure that everyone fully understands whatever new control system you implement.