Be aware that any income he receives after his death must be reported on a separate estate tax return, not his personal return.
I would expect that, as of the date of his death, you are the 100% owner of the account and all the income after that date is reported in your name. Did you call the broker? Are they able to change the account to list you as the primary owner?
For income earned prior to his death, that would be reported however you reported it before. Depending on what the account activity was, there might not actually be any income between 1/1/21 and 1/10/21 that would be attributable to your uncle. I would imagine that most investments won't pay interest or dividends during that time frame, so unless you sold something with a capital gain or loss during those 10 days, there might not be any 2021 income in your uncle's name.
I would start with the estates department at the brokerage, then contact a professional tax advisor if you are still unsure.
