Welcome Mumzy! Congrats on your little one! Do you know if it's a boy or a girl?
One project that works for babies at any time is a wee blanket - not a huge bedspread, just something that will keep them warm during naps. I made the Log Cabin blanket from Mason Dixon Knitting for a friend's baby this spring. It's a very simple pattern (all garter stitch), but it looks really nice, especially if you use several colors.