WABetaInfo shares that your message may not appear to be edited for the other person if they do not check your message thread within a certain amount of time. If a person does not check the thread within a day or so, your edited message may not appear as such.
To test this its edit button with a set timeframe for such a touch-up, Twitter is going down the route of bringing in a broader audience. However, while there is a time limit to edit a tweet, they will always appear as such. The edit button by WhatsApp is still under development, so until the newer version is downloaded, this will not be available to the beta testers.