The problem is the cache of the old Java and here are 2 ways to clear it.
First solution thanks to Revnik:
Go to Start, Run, type "javaws -viewer" and hit enter.
Highlight Wurm and hit the red X to delete it.
Click "Close" in the bottom right.
The Java Control Panel should be open. Click the "Security" tab.
Highlight "OnetooFree AB" and click "Remove".
Close everything and try running Wurm again.
Also, you can click Play on WurmOnline.com. If you are running IE you can right click and "launch application" rather than downloading a new file every time.
A second Windows version to correct it:
1. Uninstall all java installations in Add/Remove Programs
2. Go to www.java.com in your favored web browser and install the latest java update it recommends.
3. Open up the Java Control Panel in windows Control Panel ( in the Programs section of Win 7)
4. Look at the [Temporary Internet Files] section and click [View].
5. Right click on and [Delete] each Wurm client then click [Close]
6. Look again at [Temporary Internet Files] section and click [Settings]
7. Click [Delete Files] and the default settings should be already set so click [Ok]
8. Click [Ok] and close out of the java control panel and any other windows.
9. Go to wurmonline.com and click Play
This may be a step or too more involved than needed, but it works.