Before we begin, keep in mind that a total of three updates must be installed on the path to Mango: 7392, 7403, and 7720, which is Mango itself. The process will take a while. Some people may already have 7392, in which case things will move a bit faster. I also recommend removing any music and unessential apps from the phone beforehand, as that can significantly cut down on the time it takes to backup and restore the phone's data.
Steps to force a Mango update on phones for which Mango has started rolling out:
Connect your Windows Phone to a PC and launch the Zune software
Click on Phone > Settings > Update. Allow the software to tell you whether an update is available. If so, skip this guide. If not continue to the next step.
Time to force the update. Click another button (any of the options above or below the word Update). Then click the Update button again, and this time disconnect your computer from the internet after 1-2 seconds. Do this by pulling out the Ethernet cable, disabling Wi-Fi, etc. If it says your phone is already up to date, you didn't disconnect quickly enough and need to try this step again.
Wait 30 seconds or so and Zune should display a notification that an update is available. This is 7392, the first Mango pre-update.
Now reconnect to the internet and continue with the update process through the Zune software.
After 7392 is fully installed, you may receive a notification that another update is available. If so, install it now. If not, proceed to the next step.
If the second update isn't popping up on its own, repeat steps 3-5, causing the 7403 update to appear.
Install 7403, brave reader. Speeding right along, isn't it?
Mango time! At this point, Zune should automatically start installing the last update, 7720, aka Mango. If it doesn't happen automatically, force it like we did in previous steps.
This last step is very important! Celebrate your phone’s new found Mango status by purchasing a mango smoothie at your nearest Smoothie King, Jamba Juice, Baskin Robbins, or similar establishment.