If you filed one tax return listing your child as a dependent with "applied for" as the tax number, and mailed it with the W-7 at the same time, I think you have done all that you can do. Filing an amended return probably won't change anything.
I don't actually know what the IRS will do now. If they processed your tax return before the ITIN was issued and gave you a partial refund, I don't know if you will now get the rest of the refund automatically. This is a new area of tax law.
A very recent change by Congress says that a taxpayer can't get tax benefits for a dependent unless the dependent has a tax number (SSN or ITIN) at the time of a "timely filed" tax return. In other words, if you file before the deadline with no dependent, get the tax number after April 18, and file an amended return, you can't get retroactive tax refunds for the dependent.
The best advice would be to apply for the tax number first, and not file any tax return until the tax number was issued. The taxpayer can file an extension request which extends the deadline to October, so the return can still be "timely filed."
However, most taxpayers are not allowed to apply for an ITIN unless they file the application with the tax return. In this situation, you filed a timely return, but did not have an ITIN. It's not clear what the laws allows for in this situation or how the IRS will interpret the law. Some experts have suggested that as long as the ITIN was approved before April 18, the IRS should pay the tax benefits. Other experts have disagreed.
You can try and file an amended return listing the new ITIN in place of "applied for" but I don't know if it will be approved, and I suspect it won't. The IRS may process the difference on their own, but I don't know as this is a new area of tax law and this is the first year where this new law has been in place.
You may want to contact an attorney who is an expert in immigration tax laws, or look for blogs from immigration tax attorneys, who might have more information.
