That's roughly it, for non-emergency citizen's treatment basically you either call a NHS helpline for basic advice or go to your GP (General Practitioner/local doctor) and they give you a referral to an NHS specialist or your insurance company for anything they can't handle. For non-residents the full details are...
Am I entitled to NHS treatment when I visit England?
When you visit England, you’ll normally have to pay for all NHS treatment unless you’re exempt from charges. ...
Regardless of how long you’re staying or your nationality, you’re entitled to free emergency NHS treatment from:
a primary care practice, such as a GP surgery
an A&E department
an NHS walk-in centre
However, unless you’re exempt from charges, you’ll have to pay NHS charges if you’re:
admitted to hospital as an in-patient (this includes high dependency units and other emergency treatment, such as operations), or
registered at an outpatient clinic
" ...the skinny version is you get a high level of patching-up, sticking back together services and/or made stable for free. If you have to be admitted for a long duration recovery or on-going treatment then you have to start paying unless your government has a prior agreement (probably all of Europe and the commonwealth, plus a few random others I'd guess).
As far as I can tell our A&E units can treat basically any injuries and issues that arise from an accident or initial infection etc..