We run MMA fight nights and we spend a lot of money on getting very good medics, we have two paramedics who bring an ambulance with them. As soon as someone is knocked out the ref is usually the first to put the fighter in recovery position and removes his gumshield, the medics are in as fast as they can.(they aren't called in they come in automatically when any fighter is KOd.)
From that point on the medics are in charge. the fighter has no choice as to whether he receives care, we as promoters have a duty of care to make sure they receive the best medical care. I will keep an eye on all the fighters afterwards too and have often called a medic to check one out if I feel any concerns.
In MMA of course there is no standing count so even if a fighter isn't fully KOd the fight will still be stopped.
I'm not sure how our paramedics correspond with yours as in qualifications but while they are expensive to hire they are worth their weight in gold, we wouldn't use first aiders for this. They also do the medicals for the fighters before hand. Some shows use doctors but to be honest your average family doctor knows little about emergency work.
http://www.nhscareers.nhs.uk/details/Default.aspx?Id=905