I received none.

That means either a) You're account was not included in the comprised accounts or b) You checked an email different from your LinkedIn account.

I didn't get an email until after I logged into my account with the compromised password. Then it sent me an email and locked my account. Sort of odd way of dealing with the problem.

This all happened about 20 minutes ago.

It was just a coincidence that you received the email after you logged into your account. Logging into your account didn't trigger the email to be sent.