As long as you can get hold of forage, can manage to keep them supplied with clean water and can keep on top of infection, your sheep will be fine.

On the subject of infection - my advice, if you are going to be keeping young lambs in for more than 24-48 hours after birth, is to use prophylactic antibiotics with the newborns. Orojet or Spectam or similar, one squirt per lamb within one hour of birth if you can manage it. It will help fend off any outbreaks of watery mouth etc. And keep the hygiene levels up - thoroughly clean out each pen after each occupant, put lime down under the new bedding. Watch out for leakage from one pen to another - clean out pens of longstay occupants (adoptive families, mothers needing time to bond with their lamb(s), etc) every few days so you don't get a build up of any infective material.
On the subject of water, lactating ewes kept indoors need a phenomenal amount of water - some for the milk they're making and some to help them digest the dry food. I expect to need to give each ewe at least two buckets of water a day.
Space for growing families could be another problem if you have to keep the sheep in longer. Too many families in a smallish space can lead to lambs getting knocked about by other mothers. If your space is too limited, you may be better to keep everyone in individual pens most of the time, and let a few families out to a larger area for exercise (and play as the lambs grow) at a time.
Lambings like this one test our mettle. Such a shame when a one like this is your first, but they won't all be like this, Rachel.

Chin up, chick. Enjoy your lambs, whether they're indoors or out.
