I had to have my DS induced at 37 weeks, due to risk of still birth if I went longer (different problem to yours though).
They will be very careful with you during your labour and they will probably want to get the baby out relatively quickly rather than letting labour go on an on. They will have a SCBU bed ready as well for your baby, BUT he/she may not need it. I was absolutely terrified beforehand, just like you but the attitude I tried to use is that your condition has been diagnosed, they are well aware of your problem and will take all necessary precautions with both of you. I also had to prepare for a Csection as induction is often not successful early, however if they advise this for you, try to go with it and remember they are doing the best thing for you. At the end of the day, like another poster has said, the goal is to get the baby out healthy and this is what the doctors will do for you, regardless of the method of delivery.
My DS came out at 37 weeks and was fine, just he was very scrawny as he didn't have his last few weeks to put on weight. He is now 20 months, absolutely fine, but still skinny, although this is no problem and he eats well. He was also able to breastfeed when he was born, but I believe his suck was quite weak compared to a big strong term baby, but he could still feed. I just let him feed all the time and he was fine.
Try not to worry, they will look after you.