How do I find out if my unemployment tax relief refund is in process? My tax return was already filed before the announcement, an email stated no amendment needed.

There is no way to find out.   The IRS does not have a stratus page.

Refunds are being sent now and will continue in batches over the next 2-3 months until complete.

If you already filed - do not amend. See this:

https://www.irs.gov/newsroom/irs-to-recalculate-taxes-on-unemployment-benefits-refunds-to-start-in-may