CEO & CTOI got a chance to look at this today and identified the issue. IE11 has very strict security settings which cause it to behave as if there is no Java on the machine, unless it is given permission to use it (by selecting 'Run control' at the bottom of the window when the IE11 alert pops up). It could be said that it wasn't the check itself that was at fault, in that sense. In fact, we got it straight from the Java website, as their recommended method!
Once IE11 had permission to use Java, then the update check would function correctly and not pop up. But the update popup has caused more problems than it has resolved in general, particularly when two versions of Java are installed, as Andrew pointed out. We'll have to figure out an alternative solution for notifying people of the need to update their Java version.
Could you try it again and let me know if it is working as expected in IE11 now? I've disabled the update alert/prompt until further notice and will consider an alternative route for notifications.