I'd recommend going to the hospital rather than a walk-in clinic, but that's probably my own prejudices about walk-in clinics being annoying! The hospital would probably be more approachable.

As far as I'm aware, you can attend both. You can't obviously go to a GP (but why would you?)

A&E *tends* to have longer waiting times for non-emergencies, but honestly... it's hugely variable. Perhaps pick a nice sunny day (ha) and the emergency cases tend to go down. ;)