Are you eligible for childcare rebate and/or child care benefit? I'm pretty sure you can claim it with a nanny. No idea how much, but might be worth looking into?

Otherwise I'm certain there are centres that will be safe for your boy. Every centre I've worked in, despite all their other failings , have been very strict on allergy management.

Good luck I hope you can find a solution you are comfortable with.,
You can only claim CCB, CCR through a registered provider - such as LDC, FDC - most nannies aren't registered providers.