I lived in New Zealand for 10 years where healthcare is heavily subsidised but not necessarily free. Public hospitals are free but GPs and other treatment providers can charge what they like. Some charge nothing, some charge a lot, most charge a modest fee. You can also choose your GP, it doesn't matter where you live, as long as they're not overbooked they will take you. For example, the GP I went to was in the central city and she charged around $40 (about £15) for a consultation, which I thought was a very reasonable amount. It was very rare not to be able to see her at the time I wanted either, although her practice was a very busy one.
I wouldn't mind paying a small fee to see the GP here if it meant I could see the doctor I wanted to see and not just a random person within a large practice. Also, two of my friends in NZ have had babies in recent years, they both had an independent midwife. One friend paid nothing, the other had to pay a small amount to hers. It's not income related or anything like that, it is up to you to choose who you see, so if one doctor is too expensive, you just choose another. It works pretty well, but there are still similar problems to the NHS with the public hospitals being overburdened.