It's good that you are trying to make your verb tenses agree. A couple of suggestions to offer:
1. We (NA) don't have a nap, we take a nap.
2. To say "we start taking a nap" indicates that we implement the practice of taking a nap- we decide that we will take a nap every day at the same time. If you mean to talk about one particular nap (as clearly you are), then write it in a different way:
'When one or more of us starts a nap in the afternoon, he goes out to the bathroom and sleeps there.' OR
"When we take an afternoon nap, he will go out to the bathroom and sleep there.'
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