I don't know how British speaker pronounces their /r/ sound.
But If I'm not mistaken, American pronounce the /r/ in the palatal approximant way.
to make the /r/ sound, round your lips and make the er sound. Do not let your tip of the tongue touches the hard palate. Your tip of the tongue would move toward the beginning of the hard palate, but it doesn't touch any part of your mouth. It takes time to practise. I wasn't able to do so, But I got a hang of it after practising for ages. Give it a shot, dude!