where is wordpress search pagination

The code is in wp-includes/general-template.php.

However there aren’t any great options for styling the output: there’s a hook for you to modify the link URL, but nothing else. Your best bet might be to call it in type=array mode

$links = pagination_links([ "type" => "array" ]);

which will return you an array of strings of links of the form

<a class="prev page-numbers" href="https://wordpress.stackexchange.com/page/1/">&laquo; Previous</a>
<a class="page-numbers" href="https://wordpress.stackexchange.com/page/1/">1</a>
<span aria-current="page" class="page-numbers current">2</span>
<a class="page-numbers" href="http://wordpress.stackexchange.com/page/3/">3</a>
<a class="next page-numbers" href="http://wordpress.stackexchange.com/page/3/">Next &raquo;</a>

which you can then process into Bootstrap form:

  1. replace “page-numbers” in the class with “page-link”
  2. convert the current page span into a regular link with class=”active” too
  3. wrap each item with in <li class="page-item">...</li>
  4. and then wrap the whole thing in <ul class="pagination>..</ul>.

However it may be simpler to just copy/paste the function into your own code and change the mark-up it generates, rather than trying to unpack and rewrite mark-up after the fact. You potentially lose any future core updates to the function, but I think those are unlikely beyond occasional aria changes.