Not in my part of London (ie not the middle of nowhere!) OOH GP closes at 8am. That's it until 8am the next morning. Our choices then are one hospital with kids A&E, or another that has an urgent care centre, but all the kids get sent to paeds A&E anyway.
Unfortunately, we have been prolific users of A&E since my DD was born. Midwives sent us to A&E with feeding concerns (low birth weight, severe reflux, not feeding well,jaundice). An accident. A very high fever lasting more than five days and a fast heart rate (query kawasaki disease, bloods for sepsis done, discharged as viral and later dx with ear infection and chest infection), more severe feeding issues (sent by a private paeds Dr as he was so concerned, but not the right infrastructure in the private sector to treat paeds who are acutely unwell) and vomiting for several days with previous low blood sugar (required antisickness medication, which did the trick). Twice needed an ambulance due to choking on vomit and turning blue as a small baby. However, we've managed bronchiolitis at home as well as all the usual coughs, colds, fevers, and covid.
Sometimes, you're just unlucky and need to use A&E more than you'd like. Imagine if OP didn't take her kid on medical advice, and then they became severely unwell. She'd have social services on her doorstep! She isn't a Dr, and it's wrong that PP are trying to make her feel guilty for following advice from NHS staff!
However I will say OP, 3/4 hours is good. I've been in A&E with a small baby bought in by ambulance for choking and turning blue and not seen a Dr for EIGHT hours. I assume 24 hours away from home if we need to go to the hospital. I've also changed GP practice to one much more helpful. The last one just used to direct me to OOH when their daily appointments were taken. OOH were then inevitably full and directed me to A&E. New GP will always squeeze children in. They're actually like a GP service used to be. And only about 2 miles down the road from the old one. The difference is, they haven't been purchased by an American conglomerate hell bent on profiteering from the NHS. But it isn't a race to the bottom and the state of the NHS is not okay.