Hm, I earned two trophies in Mirror's Edge and updated but it still hasn't updated. I think this may have something to do with the fact that the trophies were already marked as earned on the website before I updated. But I am being completely honest when I say that it works properly when you view my profile through a Playstation or the app.
But to kind of verify myself a little bit for Speedlunky, I do have quite a bit of experience with Spelunky. My steam profile is https://steamcommunity.com/id/XylophonePenguin, and I changed my display name on there to my PSN name to confirm that it is me. It has nearly 100 hours played of Spelunky, and one of my Steam friends, Hectic, was a well-known Spelunky player with a very fast time, very high scores, and a large number of hours. We met through the Spelunky community, if this gives me any sort of credit. The reason that I earned the trophy for completing the Ice Caves three minutes before Speedlunky is that my intention for my first completed run was to unlock this trophy. I have already done so in the past on another platform, so it really didn't take as long as the trophy rarity would have you believe, as it is largely based on your skill and knowledge of the game. My first run that made it past the Ice Caves also made it through the Temple, the final level required to beat the game, and if I recall correctly I was somewhere around 4 minutes in when I finished the Ice Caves. Pausing and the transitions between levels both stop the in-game timer, which is used to unlock the trophy, so this means that after about 3 and a half minutes I had finished the Temple and unlocked Speedlunky.
I don't really have a way of verifying this, but I was also a Spelunky player on the Xbox 360. My profile there is https://www.trueachievements.com/gamer/Hawk+rXz, and this is where I finished my first Hell run. If you need me to verify this, I can respond to a message or something on Xbox Live, but changing my name there costs money and I haven't used my Xbox in years so I would prefer not to do that.
I am going to try hiding and then unhiding Spelunky from my PSN profile to see if this fixes it. I'm sorry this happened, honestly I did double check that syncing worked using PSN, but I never checked on the site, so I didn't realize that the timestamps were still illegitimate. In the meantime, I am going to post some screenshots of what psnprofiles is saying the timestamps are, as I believe the game will be hidden from my profile for a few minutes. (https://imgur.com/a/58tFL6p)
EDIT: I can only update my profile on psnprofiles every 60 minutes. Darn. So since my plan to private and unprivate will take two updates, and I have already updated to try to add Mirror's Edge trophies, my profile may not be sorted out for around two and a half hours.