When I called my GP about a suspected broken toe, I was told to go to A&E. A&E is accessible to all, and you do not need to show your NHS number upon arrival. I always bring some form of proof that I'm eligible for NHS treatment and ID anyway, but haven't been asked for it.
If it's still bothering you, feel free to go. It is likely to be a long wait, and there's no real treatment for it-- other than rest, NSAIDs, and it taping to another toe. (Unless an X-Ray shows it's truly mangled, and needs to be fitted back into place first). Takes ~6 weeks to heal if it is broken.