Change theme based on window size

Your code actually works for me and it is changing the theme according to the current width.

But I think you should need some tweaking something like below:

For your JavaScript:

var $ = jQuery;

$(document).ready(function() {

    var windowWidth = $(window).width();
    var windowHeight = $(window).height();

    $.ajax( {
      type: "POST",
      url: "http://example.com/wp-admin/admin-ajax.php", // Must use absolute url
      data: {
        action: 'ajax_action',
        windowWidth: windowWidth,
      },
      success: function(data, textStatus, XMLHttpRequest) {
        alert('success ' + data); // might not be useful.
      },
      error: function(XMLHttpRequest, textStatus, errorThrown) {
        alert(errorThrown);
      }
   });

});

Use absolute URL, sometimes it won’t find the admin-ajax.php file, because of incorrect handling of the relative URL.

For functions.php

function mmf_enqueue_scripts() {
  if( !is_admin() ):
    wp_enqueue_script('winsize-detect');
  endif;
}
add_action( 'wp_enqueue_scripts', 'mmf_enqueue_scripts' ); // Use `wp_enqueue_scripts` instead of `wp_print_scripts`

function ajax_action() {

    if( isset( $_POST['windowWidth'] ) ) {
        return $_POST['windowWidth'];
    }
    return 0;
}


function mmf_change_theme($theme) {

  // to make AJAX run only before this function
  add_action('wp_ajax_ajax_action', 'ajax_action');
  add_action('wp_ajax_nopriv_ajax_action', 'ajax_action');


  $width = ajax_action();
  //use '<='
  // This might be your mistake, it worked for me.

  if( $width <= 320 ) {  
    $theme="twentytwelve";
    // if I echo the width here I get: 3203203203200 (if width=320)
  }
  else {
    $theme="twentyteleven";
    // if I echo the width here I get: 7707707707700 (if width=770)
  }

  return $theme;
}

add_filter('template', 'mmf_change_theme');
add_filter('option_template', 'mmf_change_theme');
add_filter('option_stylesheet', 'mmf_change_theme');

Don’t use wp_print_scripts because its not recommended. Use wp_enqueue_scripts instead.

Make sure don’t equal it to specific width, try less than or equal to <=.

Replace == with <=