It's not always the solution, so please don't take this as me saying that the following will always work, but from my own experiences with getting stuck on loading screens, I find rebooting the client often works wonders.
At least for me, most slow and eventually stuck loading screens (most, not all, there are exceptions) are memory issues, especially on accounts with characters and banks that have lots of stuff. Not only is it the longer I play the more likely it starts slowing down and freezing, but if I do any banking, it starts happening much sooner.
I came to this conclusion for myself because sometimes I multibox, but my main account has a bank filled with stuff, characters with lots of stuff, and TR caches. My alt accounts do not have nearly as much. And even running on the same computer, my main account will have longer loading screens, and if I did any banking, really long loading screens, while the alt accounts go through quickly.
I'm not saying this IS DEFINITELY the issue you're facing, but it IS something that I have noticed and this information might be able to help, maybe.