My child is 5 weeks old and has (or will have) dual nationality. One of his passports has arrived but the British one hasn't yet. I'm having to travel and leave the UK for the time being. Is it legal for someone else to bring his British passport to me overseas when it arrives?
You can apply for UK passports when living overseas, the guidance here says that the passport is couriered to you. This surely implies that a third-party may deliver a passport for someone.
While I understand your caution, I think you need to have some documentary reason to suspect that carrying a third-party passport is illegal before being too worried.
Within the USA there are explicit instructions for how a third-party may retrieve a passport from a government office, this includes the need for a letter of authorisation. Again the implication is clear, it is understood by the authorities in the USA have no intrinsic issue with a person carrying a passport for someone else.
Being a cautious sort, I would suggest that you provide such a covering letter to explain that the person carrying the letter is doing so at your request.
If you want legal advice - talk to a lawyer - but if you are really asking if the person gets caught, will they get in trouble (this is entirely different than is it legal or not) then the answer is Yes.
Generally speaking, carrying two passports (any two passports) that do not belong to the person is cause for major alarm by the authorities.
It does not matter who the passport is for (relative, friend, brother, etc.) because immediately the issue arises that some sort of fraud is taking place or the person is complicit in the same.
Carrying passports for someone else -- especially across international borders is highly discouraged. I do not know of any kind of "evidence" that can be provided to get out of any scrutiny that will arise if discovered with passports that do not belong to you.
It would be much simpler and safer to use a document courier service (like DHL) to send the passport over.