The main way is via your allocated General Practitioner (GP). To see a GP you need to telephone the GP's place of work, known as a surgery, and to have any chance of getting a same day appointment you need to call at 8am precisely. Don't call at 7.59 because you won't be connected, and don't leave it until 8.01 because it will be far too late. Call within the first few seconds after 8am and you'll be informed that you're in a queue, behind several dozen earlier callers.
After an hour of waiting you'll get through to a non-medical receptionist who will ask you about your symptoms, and make a judgement about whether or not you or your child is worthy of one of the few remaining appointments for that day. If you're a very pushy mother, or someone who really enjoys talking with GPs' receptionists (which most people don't) then you may have some success in this. If you're just a chilled out guy who doesn't like to fuss, you've got no chance.
You'll then be informed that to make an appointment for a later date you need to try again for a same-day appointment the next day by repeating the process above the following morning. It is not possible to arrange an appointment for any time further in the future no matter how long you are prepared to wait.
The only other means of accessing the NHS is by rocking up at the Accident and Emergency (A&E) department of your nearest large enough hospital. Unsurprisingly given the above, these are rather busy and you may be waiting for a day or so before you get seen. To avoid adding to this delay you should make you own way to A&E because ambulances routinely take several hours to reach you for the most urgent cases (heart attacks, stokes etc) or up to a few days for, say, a fallen elderly person, and on reaching the hospital you wont be able to leave the ambulance and enter the A&E department for up to another day.
However, once you're in the system and being seen by a specialist then by all accounts you do get fairly well looked after, so putting in the hours at A&E or regularly playing the GP telephone lottery may well prove worth your while if you wish to make use of our envy of the world NHS.