var $this = $(this);
    var $userName = $this.find('#userNameLogin').val();
    var $password = $this.find('#passwordLogin').val();
    // var $_nonce = $('meta[name="_nonce"]').attr('content');
    var $_nonce = $this.find('#nonceLogin').val();
    var $rememberMe = $this.find('#rememberMe').prop('checked');
    jQuery.ajax({
        url: customData.ajax_url,
        type: 'post',
        dataType: 'json',
        data: {
            action: 'loginToSite',
            userName: $userName,
            password: $password,
            rememberMe: $rememberMe,
            _nonce: $_nonce
        },
        success: function (response) {
            if (response.success) {
                $('.loginRegisterResponse').fadeIn().addClass('successToLogin').find('p').text(response.msg);
                let delay = 3000;
                setTimeout(function () {
                    window.location = response.redirect;
                }, delay);
            } else {
                $('.loginRegisterResponse').fadeIn().find('p').text(response.msg);
            }
        },
        error: function (error) {

        }
    });
});

$user = wp_authenticate_username_password(null, $userName, $password);
‎

$creds = array();
    $creds['user_login'] = $userName;
    $creds['user_password'] = $password;
$creds['remember'] = true;

    $userLoggedIn = wp_signon($creds, false);
    if (is_wp_error($userLoggedIn)) {
        $success = false;

        if (!$success) {
            $response['success'] = $success;
            $response['msg'] = $userLoggedIn->get_error_message();
        }
        wp_die(json_encode($response));
    }

    if ($success) {
        $response['success'] = $success;
        $response['redirect'] = home_url();
        $response['msg'] = __("hi ", THEME_NAME) . $userName . __(' , you signed in to site successfully', THEME_NAME);
    }
    wp_die(json_encode($response));
}

<form id="loginFormSubmit" class="forms_form" action="<?php echo get_the_permalink() ?>"
                              method="post">
                            <fieldset class="forms_fieldset">
                                <div class="forms_field">
                                    <input class="forms_field-input" id="userNameLogin" type="text" name="userName"
                                           placeholder="<?php echo __('username', THEME_NAME); ?>" required
                                           autofocus>
                                </div>
                                <div class="forms_field">
                                <input id="passwordLogin" type="password" class="forms_field-input"
                                           name="password" placeholder="<?php echo __('password', THEME_NAME); ?>"
                                           required>
                                </div>
                                <div class="col-auto my-1">
                                    <div class="custom-control custom-checkbox mr-sm-2">
                                        <input type="checkbox" id="rememberMe" class="custom-control-input">
                                        <label class="custom-control-label"
                                               for="rememberMe"><?php echo __('Keep me Signed in', THEME_NAME); ?></label>
                                    </div>
                                </div>
                            </fieldset>
                            <div class="forms_buttons">
                                <button type="button" class="forms_buttons-forgot">
                                    <?php echo __('Forgot password?', THEME_NAME); ?>
                                </button>
                                    <input type="submit" value="<?php echo __('Log In', THEME_NAME); ?>"
                                           class="forms_buttons-action">
                                </div>
                                <div class="group loginRegisterResponse">
                                    <p></p>
                                </div>
                                <?php echo do_shortcode('[bws_google_captcha]'); ?>
                                <?php wp_nonce_field('ajax_login_form', 'nonceLogin'); ?>
                            </form>