<ul>
  <li>''Email'': {{!!email}}</li>
  <li>''Phone'': {{!!phone}}</li>
  <li>
    ''Family'':
    <ul>
      <$list filter={{!!family}}>
        <li>{{!!title}}: {{!!phone}}</li>
      </$list>
    </ul>
  </li>
  <li>''Manager'': {{!!manager}}</li>
</ul>

Here's the same thing, retaining the outer bulleted list in wikitext. The critical part is that the <ul> has to be on the same line as the end of the Family bullet – otherwise it won't be treated as a nested list.

* ''Email'': {{!!email}}
* ''Phone'': {{!!phone}}
* ''Family'': <ul>
  <$list filter={{!!family}}>
    <li>{{!!title}}: {{!!phone}}</li>
  </$list>
</ul>
* ''Manager'': {{!!manager}}