The IRS takes care of your federal tax return---not your Massachusetts return. They are completely separate taxing authorities. TurboTax does not get any information from the IRS or state after you file your tax return.
The IRS began to send out the additional refund checks for tax withheld from unemployment in May. The refunds are being sent out in batches—starting with the simplest returns first. If you are eligible for the extra refund for federal tax that was withheld from your unemployment the IRS will be sending you an additional refund sometime during the next several months. TurboTax cannot track or predict when it will be sent. The IRS has not provided a way for you to track it, so all you can do is wait for the refund to arrive.
Please read this very recent news from the IRS:
Since you are a resident you should visit the MA Dept of Revenue website where you will find the following:
"Since the Massachusetts income threshold is different from the federal income threshold, some taxpayers may be eligible for a deduction on their federal tax return but not on their Massachusetts tax return."