Thanks, Joe. I know it gets into the Yahoo webmail quickly because I checked today for that very thing. I don't want to pay them any money. If it's really important that I get something quickly, I will just go to the webmail first. It's irritating though.
I once tried to sign up for Gmail and they asked way to many questions like Google does if you want an account, so I steered clear of accounts with either one of them.
Forwarding email means sending it to another account, not accessing the mail via different client software as you are. Without unfettered access to Yahoo's coding there is no way of accurately determining what, if anything, is going on. It would not be the first time Yahoo messed with their free email accounts to encourage users to go through Yahoo's online webmail client and expose them to the advertising and marketing Yahoo relies on for their financial survival. (At one time the webmail client was the only way to use a free Yahoo email account.) It should be noted Gmail is, if anything worse than Yahoo, just sneakier, in selling your email data and content to advertisers. Nothing goes through Google that isn't available to advertisers for a price. (If you believe the claims the data cannot be linked to you as an individual I would like to talk to you about an investment opportunity in a bridge in Brooklyn.)
If you want to test to see if Yahoo is delaying sending your email try logging onto their webmail portal and see how much delay you see between a given message showing up on their webmail and in Mail on your computer. If you find a significant difference consider paying for your Yahoo account.
NOTE: As a reality check no one provides truly free email accounts. You will either pay for it directly in the form of fee for service or indirectly in the form of being bombarded with advertising, spam, and significant loss of privacy.