iCloud email delivery is very delayed

Some emails to my iCloud email account are delayed for over a day. Duplicates of those emails are delivered immediately to my gmail account. Extremely disturbing since the emails are notifications of new COVID vaccine appointments from the local health dept. which fill up very rapidly. Unfortunately, the Apple mail app does not allow push notifications from gmail, hence the reliance on using a icloud email account. Using ipad os 14.4 and Apple's Mail app. FWIW, after Apple's previous misadventures with offering email accounts, I was hoping their third attempt would be better.