I had my baby at 35+3 and he was just fine.
He weighed 4lbs 10oz (he was long but quite skinny) and we both stayed in the hosp for 5 days. He needed a hot cot to help him keep his temp up at first and when we got him home I had him bundled up in hats and booties and mits for a couple of weeks.
He was on the cusp of needing treatment for jaundice (they had to do the blood test to confirm after the skin test) but in the end he didn't need it.
We didn't need any help from the special unit at all and he had a natural birth - although I had to do the whole thing hooked up to the baby heart monitor which was a bit of a pain. I assume you're being induced? The midwife told me that early inductions are often quite fast (compared to late term ones) as your body is aware that something isn't quite right and is already getting itself into "go mode". I had the gel at 11am, got a sweep and sent round to the labour suite at 6pm and he was here at 9pm.
I did have an issue breastfeeding as my body just wasn't ready, he couldn't suck properly and my milk simply didn't come in - but I don't know how common that is, it might have just been me. Either way, he was (and is) very happy on Aptamil and it made me feel better being able to see how much he was taking in.
He's now ten weeks and on track with developments for his adjusted age and has had no problems at all thus far.
Hope this is reassuring xxx Good luck xxx