I did ask about Gluten etc then and they said the only way to tell was through an elimination diet.
If they didn't mention having the Coeliac screening blood test, then I would be asking the naturopath to refund your money, because they are clearly lacking in the knowledge and expertise and have given you misleading advice.
Can I ask what "tests" they did on the day to determine if your child had issues with those foods?
Unfortunately there are some dodgy naturopaths out there but there are also some brilliant ones. We took DD1 to see one many years ago, and the first thing she mentioned was gluten - but she insisted we go to our GP for the screening blood test before she'd 'treat' our DD - as she wanted to make sure that Coeliac disease was ruled out.
I would be taking your child back to the GP and ask for the coeliac screening blood test. Doesn't hurt for them to test for iron as well (this tends to be low in those with Coeliac disease).
Your GP can also order tests on a stool sample.
I would then ask the GP for a referral to a dietician - as they are more qualified to help you work out food issues, rather than a naturopath.
Also ask for a referral to a paed gastro.