At my GP surgery it is standard practice to ring up the morning you need an appointment and book a time slot for that day.
The lines open at 08.30 and by 08.45 thug appointments are usually all gone.
There are are pre-bookable appointments in the morning (between 8.30-9.00) but the waiting lists for these appointments are usually 3 weeks.
There is a system though where you can ring up between 2.00-2.30pm to see if there are any cancellations for the afternoon or evening appointments and if so you can book yourself in.
There is always a set doctor on each day who solely carries out home visits and phone consultations so that's really helpful too.
There are also two advanced nurse practitioners who can do about 75% of what the GP does and it's pretty easy to get an appointment with them if the GP slots are filled up.
I have on occasion needed an emergency appointment (one for my son and once when I was pregnant) and upon speaking to the receptionist and explaining (at any time of the day) they will either get me in somewhere (usually to see the doctor who is on phone consultation duties) or they will get a GP to phone me which always happens within about 30 minutes.
Regarding same day booking appointments, as far as I know all GP surgeries work this way. I used to cover the booking line at a GP surgery 8 years ago and this system was in place even back then.
Thankfully I've had no problem getting appointments though it is frustrating trying to book one on the morning and all you repeatedly get is the engages tone for about 10 minutes 