As long as it's a simple headscarf that won't obscure your face or have bits that fly around (ie there's nothing hanging loose), it shouldn't be an issue in most places.
I wouldn't wear casual clothes if this is your work experience (if this was a job, nobody would really care) - there may be times where you're not in theatres and on a ward (the hospital/trust may also have a policy on theatre blues not to be worn outside of theatres/ITU). It'll look pretty bad on you if your shadowing a doctor and you're wearing jeans.