Are you using Gmail or Google Workspace accounts and encountering these issues? Google-managed accounts archive deleted messages by default instead of actually moving them to trash. You'll need to change the setting in iOS first to make the Google-managed accounts in the Watch Mail app delete a message instead of archive it.
On your iPhone, go to Settings > Mail > Accounts > [account name] > [account name] > Advanced. Under "Move discarded messages into" select "Deleted Mailbox". The Trash swipe action should now show up both on the iOS and watchOS Mail app.
If you still can't use this properly, it might take a few minutes to sync. Otherwise, a force sync by pairing and unpairing can help as a last resort. This should fix the issue in either case. Let us know how it goes and have a great day!