mazm, you really are worrying totally unnecessarily! The fact is that your wife is totally entitled to claim Child Benefit and you are your wife jointly are entitled to claim Tax Credits. And to reinforce that, if just one of a couple living together is a sole claimant for Tax Credits, that could amount to benefit fraud, something of course to be avoided.
So why is it OK for you to be a joint claimant for Tax Credits, notwithstanding the "No Recourse to Public Funds" restriction on your visa? You need to appreciate that the definition of Public Funds in para 6 of the Immigration Rules does not cover ever possible type of benefit paid by the UK Government. And also, in view of paras 6A and 6B, it is sometimes the case that a benefit within the Public Funds definition is excluded from that definition ... for a particular claimant!
Technically ... reg 3(2), Tax Credits (Immigration) Regulations 2003 says that where one of a couple is a person not subject to immigration control .... your spouse .... and the other one is a person subject to immigration control .... you .... then
for Tax Credits purposes only both of you are treated as not subject to immigration control .... hence it is totally OK for you and your spouse to claim Tax Credits.
This is reinforced by para 6B of the Immigration Rules, which states :-
6B. A person shall not be regarded as having recourse to public funds if he is a person who is not excluded from specified benefits under section 115 of the Immigration and Asylum Act 1999 by virtue of regulations made under sub-sections (3) and (4) of that section or section 42 of the Tax Credits Act 2002.
-: and I assure you that the Tax Credits (Immigration) Regulations 2003 were indeed made under section 42 of the Tax Credits Act 2002. So mazm, from para 6B, ...... "
A person shall not be regarded as having recourse to public funds if ......" ...... your joint claim for Tax Credits is
not, for you, a claim for Public Funds.