Hey there...
Whichever client you were using, you can download it here:
http://www.uo.com/Client-Download
As for your user data... If you still have access to your old computer's hard drive and depending on your operating system:
Copy everything in these directories to your new computer ... Also depending on which operating system you are now using...
Windows XP:
C:\Documents and Settings\(Your name)\My Documents\EA Games\Ultima Online Stygian Abyss\User Data\(Both account names)\
Windows 7:
C:\Users\(Your name)\My Documents\EA Games\Ultima Online Stygian Abyss\User Data\(Both account names)\
What you should do is install the client, start it up and login with both accounts.. Then copy the files into the corresponding directories/folders on your new computer... If you are using Windows 8 now, I am not sure where the files are stored.. More than likely it's in the same directories as Windows 7, but ... You will have to check.. Or do a search on Stygian Abyss in Windows Explorer...
Hope that helps...