After some works I manage to make it works by passing a lang parameter to my WP REST API request (?lang=en for example) with this hook :
function polylang_json_api_init() {
global $polylang;
$default = pll_default_language();
$langs = pll_languages_list();
$cur_lang = $_GET['lang'];
if (!in_array($cur_lang, $langs)) {
$cur_lang = $default;
}
$polylang->curlang = $polylang->model->get_language($cur_lang);
$GLOBALS['text_direction'] = $polylang->curlang->is_rtl ? 'rtl' : 'ltr';
}
add_action('rest_api_init', 'polylang_json_api_init');