Almost surely not, but please clarify. 

What states did you live in, and for how long?

Did you earn income in either state?

Lived in indiana and moved to new york end of july 2021 and living in new york since. No income

You are not required to file a federal return because of your income level. New York would require that you file an income tax return if you filed a federal return. Since you do not have to file a federal return, you are not required to file a New York tax return. Indiana does not require a return if you had no income. 

However if I filed a federal return, is it compulsory to also file a state return for New York?

No, you do not need to file a state return for New York. 

Also wanted to know if that could delay the process of my federal tax return as its taking longer than 3 weeks. And also about the stimulus checks, since I filed for federal tax return for 2020, I still haven’t received both the federal tax return and the recent stimulus checks.

Not filing a state return will not have any effect on the federal return.

There is a delay with the IRS. Some people are receiving the stimulus payments quickly, but others are waiting quite a while.

Here is a link to check on the third stimulus payment.

This has information about the IRS delay.

So If I filed for federal tax return for 2020,  does that make me eligible for the recent stimulus checks?

Yes, the IRS will use your 2020 tax return to determine your eligibility to receive a stimulus payment.  Here is a TurboTax article about the American Rescue Plan, including the third stimulus payment.