timber: set the search query to the loaded page on the 404 page so you can extend search.twig in 404.twig.

functions.php

<?php

add_filter( 'timber/context/404', function ( array $context ) {
	global $wp;

	$url = urldecode( $wp->request );

	$context['referer'] = wp_get_referer();

	$context['url'] = site_url( $url );

	$query = new WP_Query( [ 's' => get_search_query( false ), 'paged' => get_query_var( 'paged' ) ] );

	$context['posts'] = new \Timber\PostQuery( $query );

	$context['found_posts'] = $query->found_posts;

	return $context;
} );

add_filter( 'get_search_query', function ( $query ) {
	if ( is_404() && ! get_query_var( 's', false ) === false ) {
		global $wp;

		$query = urldecode( $wp->request );

		$query = preg_replace( '@/page/[1-9]\d*/?$@', '', $query );
		$query = preg_replace( '/\.(html|htm|php|asp|aspx)$/i', '', $query );

		$query = preg_replace_callback( '@/([^/]+)(?:/|$)@', function ( $m ) {
			return ' ' . $m[1] . ' ';
		}, $query );

		var_dump( $query );
	}

	return $query;
} );