You need to set the active attribute of the inner, nested uib-tabset to null. By default, the first tab is always selected, so the alertMe function is executed.
<uib-tabset active="active">
<uib-tab index="0" heading="Static title">Static content</uib-tab>
<uib-tab index="3" select="alertMe()">
<uib-tab-heading>
<i class="glyphicon glyphicon-bell"></i> Alert!
</uib-tab-heading>
I've got an HTML heading, and a select callback. Pretty cool!
</uib-tab>
<uib-tab heading="My issue">
<uib-tabset type="pills" active="null">
<uib-tab heading="sorry" select="alertMe()">
</uib-tab>
</uib-tabset>
</uib-tab>
</uib-tabset>
Working plunker
Feedback OP:active="-1" Worked for me .