Unfortunately, I’ve been hearing from people having trouble with these Storyline badges for a terribly long time (the third Google result for ‘pottermore explorer badge’ is this post from August 2013!). I’m afraid I still haven’t found any concrete way of fixing this, and like the navigation bar it seems to be a problem within the site itself, and totally arbitrary in who it affects :/ Some methods you can try to get it working again:
1. Clearing the cache and/or cookies of your current browser - how-tos for Chrome, Firefox, and Explorer; for Macs I believe there is a ‘Reset Safari’ option.
2. Switching to another browser entirely - Google Chrome or Mozilla Firefox are recommended
3. Wait it out - it might just be taking a while to register, so try logging out and leaving it for a few days, then returning and seeing if anything has changed.
If none of these work, you can send in a bug report to Pottermore and see if they can be of any help - log in, follow this link and scroll down to click What’s Next, then Need More Help, and from there you can write in (check the circle asking for a reply and hopefully they should get back to you within a few days).
I’m so sorry that I can’t be of more help with this! :( Best of luck to you all!