I did my taxes on January as single, because my wife had no ITIN. Then about 5 weeks ago, I did a amended tax return with H & R block as married with the ITIN application for my wife. It been five weeks since I did the amended tax. We received the ITIN on the four week. When can I expect my refund cheque for the amended tax return?
You can likely expect it in the next 11 weeks, but maybe longer if they don't process the return until after the ITIN application is accepted.
Per the IRS page Amended Return FAQ:
How long will it take to process an amended return?
A Form 1040X, Amended U.S. Individual Income Tax Return can take up to 16 weeks to process once we receive it.
Not always helpful, but you can check on the status of your amended return here: