Help to edit the Recent Posts Wordpress widget to diplay in all 3 languages at once

I'm not familiar with WPML plugin but if you have language specific categories, you can simply do that:

...<ul class="recent-english-posts"><?php    $loop = new WP_Query('cat=' . get_category_by_slug('english')->term_id . '&showposts=3');    if($loop->have_posts()): while($loop->have_posts()): $loop->the_post();?>    <li><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></li><?php endwhile; else: ?>    No English posts yet!<?php endif; ?></ul>...<ul class="recent-spanish-posts"><?php    $loop->query('cat=' . get_category_by_slug('spanish')->term_id . '&showposts=3');    if($loop->have_posts()): while($loop->have_posts()): $loop->the_post();?>    <li><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></li><?php endwhile; else: ?>    No Spanish posts yet!<?php endif; ?></ul>...<ul class="recent-espanol-posts"><?php    $loop->query('cat=' . get_category_by_slug('espanol')->term_id . '&showposts=3');    if($loop->have_posts()): while($loop->have_posts()): $loop->the_post();?>    <li><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></li><?php endwhile; else: ?>    No Espanol posts yet!<?php endif; ?></ul>...

By placing this code in your theme sidebar.php you'll hopefully done. But what if you want this as a widget? there are two solutions on my mind:

First Solution: As you previously mentioned in your question update, you can fork the core! & change the standard WordPress recent posts widget. Here you go by replacing the original widget() method of the WP_Widget_Recent_Posts class:

...function widget($args, $instance) {        $cache = wp_cache_get('widget_recent_posts', 'widget');        /* pre-saving language-specific ids for ease of use & code readability ofcourse! */        $cat_ids = array(                       'en'=>get_category_by_slug('english')->term_id,                        'sp'=>get_category_by_slug('spanish')->term_id,                        'es'=>get_category_by_slug('espanol')->term_id                       );        if ( !is_array($cache) )                $cache = array();        if ( isset($cache[$args['widget_id']]) ) {                echo $cache[$args['widget_id']];                return;        }        ob_start();        extract($args);        $title = apply_filters('widget_title', empty($instance['title']) ? __('Recent Posts') : $instance['title']);        if ( !$number = (int) $instance['number'] )                $number = 10;        else if ( $number < 1 )                $number = 1;        else if ( $number > 15 )                $number = 15;        /* recent english posts  loop */        $r = new WP_Query(array('cat' => $cat_ids['en'], 'showposts' => $number, 'nopaging' => 0, 'post_status' => 'publish', 'caller_get_posts' => 1));        if ($r->have_posts()) : ?>        <?php echo $before_widget; ?>        <?php if ( $title ) echo $before_title . $title . $after_title; ?>        <ul>        <?php  while ($r->have_posts()) : $r->the_post(); ?>        <li><a href="<?php the_permalink() ?>" title="<?php echo esc_attr(get_the_title() ? get_the_title() : get_the_ID()); ?>"><?php if ( get_the_title() ) the_title(); else the_ID(); ?> </a></li>        <?php endwhile; ?>        </ul>        /* recent spanish posts  loop */        $r->query(array('cat' => $cat_ids['sp'], 'showposts' => $number, 'nopaging' => 0, 'post_status' => 'publish', 'caller_get_posts' => 1));        if ($r->have_posts()) : ?>        <?php echo $before_widget; ?>        <?php if ( $title ) echo $before_title . $title . $after_title; ?>        <ul>        <?php  while ($r->have_posts()) : $r->the_post(); ?>        <li><a href="<?php the_permalink() ?>" title="<?php echo esc_attr(get_the_title() ? get_the_title() : get_the_ID()); ?>"><?php if ( get_the_title() ) the_title(); else the_ID(); ?> </a></li>        <?php endwhile; ?>        </ul>        /* recent espanol posts  loop */        $r->query(array('cat' => $cat_ids['es'], 'showposts' => $number, 'nopaging' => 0, 'post_status' => 'publish', 'caller_get_posts' => 1));        if ($r->have_posts()) : ?>        <?php echo $before_widget; ?>        <?php if ( $title ) echo $before_title . $title . $after_title; ?>        <ul>        <?php  while ($r->have_posts()) : $r->the_post(); ?>        <li><a href="<?php the_permalink() ?>" title="<?php echo esc_attr(get_the_title() ? get_the_title() : get_the_ID()); ?>"><?php if ( get_the_title() ) the_title(); else the_ID(); ?> </a></li>        <?php endwhile; ?>        </ul>        <?php echo $after_widget; ?><?php                wp_reset_query();  // Restore global post data stomped by the_post().        endif;        $cache[$args['widget_id']] = ob_get_flush();        wp_cache_add('widget_recent_posts', $cache, 'widget');    }...

But I do not prefer to use such a solution, changing the core is not a good idea! Also this could be a bad practice due to portability reasons while you can rewrite WordPress widgets!

The second, yet preferable solution! In your theme's functions.php place the code below:

<?php     function widget_mytheme_recent_posts(){?>    <!-- your new widget code will go there            replace this comment by the first block of code in this answer,            take care of php code blocks! --><?php                } //end of widget_mytheme_recent_posts()    if(function_exists('register_sidebar_widget'))        register_sidebar_widget(__('Recent Posts'), 'widget_mytheme_recent_posts');    /* the rest of functions.php code will go here, maybe sidebar registering! */?>

Hope it help ;)