I have this issue too. I can't read messages from desktop, and also I can't read them from mobile app.
Looks like this problem lasts for at least 2 days for other freelancers (and several hours for me)..
Any when it will be solved, or at least workarond?
Btw, it was ok 1 month ago.
I've not tried it on desktop, but on mobile i simply switched off wi-fi, and it connected to cellular network, than upwork app loads messages well.
Actually I had some issues with my internet provider for last 2 months... Sometimes it does not load some websites. And for last month I think from time to time to use vpn for all my internet connections, but still not decided.
In case someone still has this issue - as I did - what helped me was to install beta version of Upwork (this version won't even load because it says "no internet connection" - when I click "check connection", everything shows up okay though).
Then I didn't deinstall this version, but installed the regular version right away. Then everything worked all of a sudden.