See https://buddypress.org/support/topic/bp_nav_menu-not-working-on-single-group-page/

bp-custom.php

<?php
// Exit if accessed directly
defined( 'ABSPATH' ) || exit;

/**
 * This will only get the displayed group's menu items 
 * 1/ if a group is displayed
 * 2/ if the theme is containing the template 'buddypress/groups/single/home.php'
 * 3/ if the theme's template replaced <ul><?php bp_get_options_nav();?></ul> by <?php bp_nav_menu();?>
 */ 
function mat_single_group_nav_menu( $menus = array() ) {
	// Simply return the menu if not viewing a single group.
	if ( ! bp_is_group() ) {
		return $menus;
	}

	// Get the current group
	$current_group = groups_get_current_group();

	// Should never be the case...
	if ( empty( $current_group->slug ) ) {
		return array();
	}

	// Init an empty single group menus
	$single_group_menus = array();

	/** 
	 * Loop through each menu items to check that
	 * the parent property is the current group's slug
	 * before adding the menu to the single group's ones
	 */ 
	foreach ( $menus as $menu ) {
		// Do not keep user's nav elements
		if ( $current_group->slug !== $menu->parent ) {
			continue;
		}

		// Add the single group menu item
		$single_group_menus[] = $menu;
	}

	return $single_group_menus;
}
add_filter( 'bp_get_nav_menu_items', 'mat_single_group_nav_menu', 10, 1 );