IT shouldn't.
If not in the EA Games folder, within Program files, it might be in a folder named Electronic Arts.
The folder with the client files might also be named UO Stygian Abyss Classic Client, as well.
Frankly, the easiest way to find it is to right click on the shortcut icon you have for the client on your desktop, and choose "properties". The "Target" line will show the path to the directory, as part of the file name.
Next easiest will be to click on Start, click on "Search", search for the program file "client.exe", then when it finds it, right click on it and choose "Open Containing Folder"
You can then click on the button to go to a higher directory, one directory at a time, to find the actual location (be sure to write the folder name each step of the way).
And, let me repeat this for emphasis - if you are using the Enhanced client (as opposed to the 2D/Classic client), UO Assist doesn't work with the Enhanced Client.
And, lastly, if you've not tried playing yet, make sure that you actually RAN the installer to install the program on your machine. The download doesn't automatically run the installer, if you saved it to your hard drive.