She's probably stressing anyway - it's natural, and it's not helpful to bottle these things up. Talk to each other. Being there for her does not mean you have to shield her from everything. Also, try not to fall into the typical man trap (and I hate to generalise, but it does seem to be very common) of transferring all your worry and stress over the baby and the fact your life is changing by going into overdrive worrying about the finances. They probably aren't as bad as you think if you go through them slowly, and there are probably other things, more practical things which you can do to ease the burden on your wife rather than taking full and personal responsibility for supporting every last need of the family. If you had a choice between doing more hours at work for more money, or being around for your family more, most women would go for practical and emotional support every single time.
How much has your family income actually dropped by, and what kind of shortfall does that leave? (You may get a better idea of this once you get the tax credits claim back, but at the moment, work with the figures you actually have and don't count the tax credits at all. You can count child benefit though - £20.30 per week, paid 4-weekly, so £81.20 roughly once a month.) If the shortfall is small, it may just eat into your spending money, which is fine, you'll be spending less anyway. If it's larger it might be worth looking into getting a payment holiday on the mortgage or something like that. Extend your overdraft (you don't get charged if you don't use it, but if you need it, it's there). Do some food shopping on a credit card every now and again, if you have one. If you build up a small, reasonable amount of debt out of necessity, it isn't the end of the world. It's not going to be forever that your finances are lowered - STOP PANICKING! You've just had a baby. If you can't use your emergency buffers now, when can you?
Practical things - is your wife expressing breastmilk? This can be very draining, possibly more so than actually breastfeeding, since she won't have all the hormones the baby would be stimulating there to help. Make sure you're doing your fair share around the house, and if she is expressing, make sure you're doing more. Keep the fridge stocked. Make food for her so it's easy for her to eat well if she's completely shattered and can't face cooking for herself. When friends and family members say "If there's anything I can do, let me know" say "Yes actually, we don't have a lot of time in between hospital visits, if you could help out with a meal or two for the freezer that would be a great help."
Can you buy a pass for the hospital car park? DP's family have done this recently as his mum has been in hospital and they have all passed it around. Or parking a short distance away if there are free places.