Is there a dress code in British hospitals?

Yes, there is. Don't show up in German sandals. Don't dress in white only - they might mistake you for a cricket player. Men should were shirt and tie, a jacket is usually not required. Wear decent trousers and clean leather shoes. Women can wear both skirt and trousers, but again their outfit should not too casual. Jeans are a not appropriate. Don't expect clean white coats every day. Be grateful if you get a white coat of your size at all.