In the US with outpatient coverage (whether gov't, through employer or private), it works mostly like you said except patients have a copay at time of service that's (guesstimating) $3 to 40 depending on insurance (specialists will be higher copay around $70-80). Medicare doesn't have a copay up front, but may have after the fact.
Copays will be more if you go "outside your network" for some insurance programs.
If they also have prescription coverage, medications would also have a copay (subsidized as you say). Example using my own: $12 for generics, $40 for preferred brand name, non-preferred $64 -- all per 30 day supply.
Physicians can't bulk bill if I understanding you're meaning. Each encounter with each patient is a separate bill.... a whole days/week/months worth might be sent at one time and be paid with a single payment, but a physician must generate a billing slip for each patient that specifies what was done and for what reason. A lot of physician time is spent in this area as each insurance company is going to have slightly different rules although most follow Medicare's lead to a large degree.
Physicians that don't have a good grasp of how to bill aren't going to last long financially.
Closest we have in the US to what you describe at least from a patient perspective is our VA (veterans) or Indian Health services. These two programs would be almost exactly like NHS - as I understand NHS.
