As far as I know there is no user action required to facilitate the change
I did receive an eMail, a year or two ago, from Microsoft, asking if I wished to change my Live.co.uk address to an outlook.com one which I agreed to, but this was an option only.
Microsoft have or are in the process of closing down the Hotmail.com and the Live.com, (and their derivatives, (Live.co.uk Etc) sites but there is not any need to change any of the actual eMail addresses.
Way I understand it with mine, where I did opt to change to Outlook.com, eMails sent to either my Hotmail or live.co.uk accounts( both the same account that had changed around 2011), were just re-directed to my new Outlook.com account.
If I replied this would show as being sent from my outlook account thus informing the sender of the change (they could use either address so it's not a decision that they have to make anyway).
At some time in the future this System may be applied to those accounts like yours where the old address is still used but I have not heard any talk of this so it will probably never happen.
I still see @Hotmail addresses on small company Vans and paperwork so that is a factor that needs to be considered by Microsoft