You're in a difficult position in applying for a visa of any kind because you are here as a visitor and to then apply for a visa the UKBA would consider your original visitors visa application to be dishonest.
Your application (for any visa) would probably be rejected on this basis and you would have to appeal.
If the COA does not come in time then you can overstay and wait until it does, then get married and then apply for a spouses visa (be advised of the criteria for a spouses visa here:- ). This would be rejected probably and you would have to appeal citing article 7 Human RIghts act (right to family life). This can be successful but to offer the greatest chance of success you should instruct a solicitor to act on your behalf and their fees could in reality not be too far off the cost of returning home and making an application from there.
Your other options is to leave before your visa expires and then apply for a fiance visa (note the criteria on the same page as above), this would get around the problem of the overstay but it would mean that you would have to make a further application once you a re here and married for a spouses visa.
Alternatively you could return home, with your wife and child, marry in the US and then apply for a spouses visa from there. These applications tend to get rejected a little bit less than fiance visa applications.
If you overstay I'm afraid you accept the risk that you will be relying on article 8 grounds on appeal to be successful.
Were the COA to come in time and you married and applied for a spouses visa (in time) then they may still reject the applications on the basis of your dishonest application for a visitor's visa and would again have to rely on art 8 ground.
The only way to stay within the rules and attempt to avoid an appealed application is to return home and make an application from there. You can make the application from here but by doing so would have to accept the risk as above explained.
I am sorry it could not be better news.
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