Some amended returns were taking over 16 weeks back in 2019, then with the mail backlog from 2020, unless the return gets lucky and winds up on the top of a stack somewhere, Id be surprised to see anything about it until the end of summer.

I saw another preparer in a different group say one of their amended returns they mailed in, the client got  a letter letting them know the IRS has it, but not to expect it to be processed until the end of November.

I did an amendment for a dependent of a client that had filed on their own so the amendment had to be mailed in.  She mailed it on March 12 and received a notice from the IRS dated May 31 that the changes requested had been made.

More precisely, it depends on which pile they put you in.  There is an initial screening by a human (well, OK, an IRS clerk) who decides whether the claim gets fast-tracked or referred to exam.  This happens with both paper and e-filed returns, but I suspect many e-filers expect the process to be much faster.