If they think it's pyloric stenosis, why haven't they sent you up to paediatrics?
Pyloric stenosis is a thickening in the muscle surrounding the exit from the stomach to the rest of the gut. It tends to show up at around 4 to 6 weeks of age, classically in first born boys. When waves of pushing hit the stomach, the milk can't go down, so it comes shooting back up. You tend to see very forceful vomiting (classically hitting the wall, rather than the floor) very soon after each feed. They're then hungry and want to feed again. If left untreated things get worse, as the muscle progressively gets more over thickened, so the baby starts getting dehydrated, and the balance of chemicals in the blood goes out of kilter.
Assessment involves someone who knows what they're doing examining a baby during/after a feed, to look for waves of pushing rippling across the abdomen, feeling for a tiny lump in the upper abdomen, and seeing the baby vomit. Blood tests can show the dehydration/chemical imbalances. An ultrasound scan can actually measure the muscle at the exit of the stomach, to see if it's too big.
Treatment, if it is pyloric stenosis, involves starting the baby on an IV drip, and using it to treat the dehydration and imbalances. The baby would be not fed, and a nasogastric tube inserted, to remove and measure stomach acid that can't drain into the gut, so they know how much of what fluid to add to the drip. The surgery, thankfully, is relatively minor.
Obviously can't tell whether or not it's pyloric stenosis over the internet. Runny poos suggest the possibility of an infection, or of cows milk protein intolerance. But a 4 week old baby with lots of vomiting, and loose poos, could dehydrate very quickly. I agree that if things aren't improving quickly, then an A&E trip is likely to be necessary.