The women who started La Leche League were all Catholic. The church supports breastfeeding and nursing during church. There is beautiful imagery in the Bible about God as a nursing mother nourishing and comforting his children. There are also numerous verses about nursing babies being present at assembies of Jewish worship. Heck, look at all the gorgeous Madonna & Child art from the middle ages that show Mary nursing with an exposed breast.
I have nursed both my kids everywhere since they were a few days old - it is so much easier to not worry about packing a bottle or scheduling when they could eat.
Nursing is not something that needs to be hidden or worked around. Use a cover if it helps you feel more comfortable - I used them a lot when my babies were newborns and before they could nurse discreetly without a cover (and I practiced in front of a mirror before going in public without a cover to make sure nothing showed).
I would absolutely nurse in the cry room - doesn't matter who is in there, that room is designed for babies and children and meeting their needs. If you wouldn't feel embarrassed to feed a baby a bottle in there, then you shouldn't be embarrassed to breastfeed in there - babies are noisy when drinking bottles too, and a bottle is an artificial nipple meant to replicate the breast (i think of them as breast dildo's ).
I personally would feel comfortable nursing during Mass - I nursed during church in the sanctuary when my daughter was a newborn. But once she got older and noiser I would go to the back to feed her instead.