Got the first child tax credit but no refund

They can base the advance CTC on either the 2019 or 2020 processed returns  so the CTC could have come first.

Your situation is not unusual at all.  Millions of people are in the same boat this year.  Long delays at the IRS are due to Pandemic-induced staff shortages, large backlog of unprocessed return, extra duties of providing the stimulus payments, last minute changes in tax laws, etc.   Delays are even longer with mailed returns or anything that requires any manual processing or additional investigation.  Of course, there are some things you should check:

If you efiled, double-check that your efiled return was indeed accepted.  If you used Online TurboTax, log in and go to the Tax Home and check the efile status, then use the tool below.

Here are some tips when using the IRS "Where's My Refund" tool:

https://www.irs.gov/Refunds

• Be sure you enter the correct SSN and filing status.
• Be sure you are using only the Federal refund amount.   Do not include any state refund or any total refund or net refund (Fed and State) that TurboTax may have provided on a screen.  
• Also be sure you are using the Federal amount before any TurboTax fees are taken out (if you chose that option.)
• Look at your actual Federal return to get the Federal refund amount, i.e., your Form 1040, Line 35a.

• You can also try phoning the IRS refund hotline.  I've seen it work for some users when the WMR tool didn't.  800-829-1954

As of June 30th there were 35 million US income tax returns that had not been processed and estimated only 3% of phone calls were being answered. Verify that here 

Could there be a hangup because my address changed in 2021? I used my new address on my 2020 return but if it changed from previous years could they flag it? Also in 2020 I received unemployment payments which I heard can cause a delay.

Once your return is processed the new address will be in the system however the unemployment exclusion is definitely holding up the return. Nothing you can do to speed up the process and if you didn't get all of the advance CTC payments you are entitled  to they will simply be added to the CTC on the 2021 return you file next year. 

What do you mean "if I didn't get all the CTC payments"? Won't it still be sent to me as soon as they process my return?

I was referencing the  2021  ADVANCE CTC payments ... if you miss getting a couple (or all) payments in the monthly advance payments you will need to get it on the 2021 return as a credit.

Ok. I was talking about my main refund that was calculated by turbotax when I filed, not the 2021 advance payments. It should still come in one payment right?

If your tax return  refund has the CTC or ACTC on it  then it is  included in the total refund and should come as one payment.