In Office 365, navigate to the Admin center, then click Active users; make sure you have "Global administrator" privileges in order to introduce the changes.
Next, you will need to click your user name in the list, and find the row "Display name", then click "edit". Here's where you change it.
In Woodpecker, go to Settings → Email Accounts, then click the account in question.
Click the refresh button next to your "From Name" to have it updated.
Remember that it might take more than an hour for Office 365 to apply the new settings.
In case it doesn't update, give it a bit more time and try again.