We spend time in a one bed flat in London. Here is what I have as essentials:
A washing machine!!
3 Sleepsuits. one on, one spare, one in the wash. With integral scratchmits. I like Mothercare ones.
2 cardigans
3 vests.
All in one outdoors thingy or a little coat.
Hat
Bibs- drooling starts in earnest at about 12 weeks- they're essential to keep number of sleepsuits down.
2 blankets suitable for swaddling.
2 sleeping bags for when your DC is a bit older perhaps 8 weeks.
Cheeky Wipes (google it- better than cotton wool and water, kinder on skin than commercial wipes).
Changing mat with little pillow.
Changing bag featuring changing mat, pockets for nappies etc.
6x muslins but tea towels are fine too. Good for catching bodily fluids!
We use an Amby Nest for DD to sleep in and its portable but if you're stuck for space co-sleeping may be a good option.
Bouncy chair- Fisher Price do a good vibrating one.
Nappies
Zinc and Castor Oil cream for bottom.
Nice to have:
Bath seat £14.99 from Mothercare and fits into Amby Nest bag, but you can bathe with your DC. My DD loves the bath and hated the sink. She used an Ikea baby bath until 15 weeks when she became too long for it.
Bottles, Steriliser and breast pump. Totally not essential if Bfing is working out for you.
hth 