"So if your baby is going pale when asleep, call NHS direct!!!"
No, go straight to A&E as fast as you can! Any colour and/or breathing changes in a baby/young child should be treated as an emergency.
Our baby was in PICU overnight aged 5 weeks with breathing problems. Although we got him there pretty quickly (we did not bother with a child seat, my wife held him upright and I drove fast), his oxygen saturation was 72% and he was not controlling his blood gases. He was quickly admitted to PICU and put on CPAP. Luckily, within 24 hours, he was on the mend, and out of hospital in 72 hours. I don't know what would have happened had we stopped to call NHS direct.
OP, best of luck with your baby, it sounds like she is getting better. The good thing with young babies is that, although they go downhill fast, they get better fast too.